Consequences of Solvability by Radicals Bernd Schr oder logo1 - - PowerPoint PPT Presentation

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Consequences of Solvability by Radicals

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Introduction

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Introduction

• 1. Galois groups are isomorphic to subgroups of Sn

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Introduction

• 1. Galois groups are isomorphic to subgroups of Sn and some
• f them are equal to Sn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Introduction

• 1. Galois groups are isomorphic to subgroups of Sn and some
• f them are equal to Sn.
• 2. It does not look like Sn has many normal subgroups.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Introduction

• 1. Galois groups are isomorphic to subgroups of Sn and some
• f them are equal to Sn.
• 2. It does not look like Sn has many normal subgroups. And

if An in an indication, there are not many ways for normal subgroups of Sn to have normal subgroups.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Introduction

• 1. Galois groups are isomorphic to subgroups of Sn and some
• f them are equal to Sn.
• 2. It does not look like Sn has many normal subgroups. And

if An in an indication, there are not many ways for normal subgroups of Sn to have normal subgroups.

• 3. So root towers for normal extensions cannot be terribly

tall.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Introduction

• 1. Galois groups are isomorphic to subgroups of Sn and some
• f them are equal to Sn.
• 2. It does not look like Sn has many normal subgroups. And

if An in an indication, there are not many ways for normal subgroups of Sn to have normal subgroups.

• 3. So root towers for normal extensions cannot be terribly

tall.

• 4. But they need not be, because splitting fields can be

generated by adjoining a single element.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Introduction

• 1. Galois groups are isomorphic to subgroups of Sn and some
• f them are equal to Sn.
• 2. It does not look like Sn has many normal subgroups. And

if An in an indication, there are not many ways for normal subgroups of Sn to have normal subgroups.

• 3. So root towers for normal extensions cannot be terribly

tall.

• 4. But they need not be, because splitting fields can be

generated by adjoining a single element.

• 5. So we still need some more ideas.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. An nth root of unity r is called a primitive nth root
• f unity iff every other nth root of unity is a power of r.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. An nth root of unity r is called a primitive nth root
• f unity iff every other nth root of unity is a power of r.

Roots of unity can be adjoined before we start, if necessary.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. An nth root of unity r is called a primitive nth root
• f unity iff every other nth root of unity is a power of r.

Roots of unity can be adjoined before we start, if necessary. From here on, demanding characteristic 0 is quite common.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. An nth root of unity r is called a primitive nth root
• f unity iff every other nth root of unity is a power of r.

Roots of unity can be adjoined before we start, if necessary. From here on, demanding characteristic 0 is quite common. Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. An nth root of unity r is called a primitive nth root
• f unity iff every other nth root of unity is a power of r.

Roots of unity can be adjoined before we start, if necessary. From here on, demanding characteristic 0 is quite common.

• Theorem. Let F be a field of characteristic 0 that contains all

nth roots of unity, let f ∈ F and let w be a root of xn −f = 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. An nth root of unity r is called a primitive nth root
• f unity iff every other nth root of unity is a power of r.

Roots of unity can be adjoined before we start, if necessary. From here on, demanding characteristic 0 is quite common.

• Theorem. Let F be a field of characteristic 0 that contains all

nth roots of unity, let f ∈ F and let w be a root of xn −f = 0. Then F(w) is the splitting field of the polynomial xn −f.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. An nth root of unity r is called a primitive nth root
• f unity iff every other nth root of unity is a power of r.

Roots of unity can be adjoined before we start, if necessary. From here on, demanding characteristic 0 is quite common.

• Theorem. Let F be a field of characteristic 0 that contains all

nth roots of unity, let f ∈ F and let w be a root of xn −f = 0. Then F(w) is the splitting field of the polynomial xn −f. Moreover, the Galois group G

• F(w)/F
• is isomorphic to a

subgroup of Zn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. An nth root of unity r is called a primitive nth root
• f unity iff every other nth root of unity is a power of r.

Roots of unity can be adjoined before we start, if necessary. From here on, demanding characteristic 0 is quite common.

• Theorem. Let F be a field of characteristic 0 that contains all

nth roots of unity, let f ∈ F and let w be a root of xn −f = 0. Then F(w) is the splitting field of the polynomial xn −f. Moreover, the Galois group G

• F(w)/F
• is isomorphic to a

subgroup of Zn. In particular, the Galois group G

• F(w)/F
• is

commutative.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k and hence Φ(µ ◦σ)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k and hence Φ(µ ◦σ) = [j+k]n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k and hence Φ(µ ◦σ) = [j+k]n = [j]n +[k]n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k and hence Φ(µ ◦σ) = [j+k]n = [j]n +[k]n = Φ(µ)+Φ(σ).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k and hence Φ(µ ◦σ) = [j+k]n = [j]n +[k]n = Φ(µ)+Φ(σ). Moreover, Φ is injective

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k and hence Φ(µ ◦σ) = [j+k]n = [j]n +[k]n = Φ(µ)+Φ(σ). Moreover, Φ is injective, because Φ(σ) = [0]n implies σ(w) = w

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k and hence Φ(µ ◦σ) = [j+k]n = [j]n +[k]n = Φ(µ)+Φ(σ). Moreover, Φ is injective, because Φ(σ) = [0]n implies σ(w) = w, and hence σ = id

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k and hence Φ(µ ◦σ) = [j+k]n = [j]n +[k]n = Φ(µ)+Φ(σ). Moreover, Φ is injective, because Φ(σ) = [0]n implies σ(w) = w, and hence σ = id (rest: good exercise).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k and hence Φ(µ ◦σ) = [j+k]n = [j]n +[k]n = Φ(µ)+Φ(σ). Moreover, Φ is injective, because Φ(σ) = [0]n implies σ(w) = w, and hence σ = id (rest: good exercise). Hence G

• F(w)/F
• is isomorphic to a subgroup of Zn

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k and hence Φ(µ ◦σ) = [j+k]n = [j]n +[k]n = Φ(µ)+Φ(σ). Moreover, Φ is injective, because Φ(σ) = [0]n implies σ(w) = w, and hence σ = id (rest: good exercise). Hence G

• F(w)/F
• is isomorphic to a subgroup of Zn, and

because Zn is commutative, so are its subgroups.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let r be a primitive nth root of unity. Then

1,r,r2,...,rn−1 are the n distinct nth roots of unity and w,rw,r2w,...,rn−1w are the n distinct roots of xn −f. Therefore F(w) is the splitting field of the polynomial xn −f. Let σ ∈ G

• F(w)/F
• . Then σ(w) = wrk for some k ∈ Z, which

is unique modulo n. Define Φ : G

• F(w)/F
• → Zn by

Φ(σ) := [k]n, where k is the unique element of {0,...,n−1} so that σ(w) = wrk. Let σ,µ ∈ G

• F(w)/F
• and let µ(w) = wrj, σ(w) = wrk. Then

µ ◦σ(w) = µ

• wrk

= µ(w)µ

• rk

= µ(w)rk = wrjrk = wrj+k and hence Φ(µ ◦σ) = [j+k]n = [j]n +[k]n = Φ(µ)+Φ(σ). Moreover, Φ is injective, because Φ(σ) = [0]n implies σ(w) = w, and hence σ = id (rest: good exercise). Hence G

• F(w)/F
• is isomorphic to a subgroup of Zn, and

because Zn is commutative, so are its subgroups.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Definition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. Let G be a finite group.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. Let G be a finite group. Then G is called solvable iff

there is a chain of subgroups {e} = G0 ⊳G1 ⊳···⊳Gm = G so that Gj/Gj−1 is commutative for j = 1,...,m.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. Let G be a finite group. Then G is called solvable iff

there is a chain of subgroups {e} = G0 ⊳G1 ⊳···⊳Gm = G so that Gj/Gj−1 is commutative for j = 1,...,m. Lemma.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. Let G be a finite group. Then G is called solvable iff

there is a chain of subgroups {e} = G0 ⊳G1 ⊳···⊳Gm = G so that Gj/Gj−1 is commutative for j = 1,...,m.

• Lemma. Let G be a finite group and let N ⊳G.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Definition. Let G be a finite group. Then G is called solvable iff

there is a chain of subgroups {e} = G0 ⊳G1 ⊳···⊳Gm = G so that Gj/Gj−1 is commutative for j = 1,...,m.

• Lemma. Let G be a finite group and let N ⊳G. If G is solvable,

then G/N is solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = / 0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj. Gj−1N/N is normal in GjN/N:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj. Gj−1N/N is normal in GjN/N: Gj−1N/N is a subgroup of GjN/N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj. Gj−1N/N is normal in GjN/N: Gj−1N/N is a subgroup of GjN/N. Let gjN ∈ GjN/N and let gj−1N ∈ Gj−1N/N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj. Gj−1N/N is normal in GjN/N: Gj−1N/N is a subgroup of GjN/N. Let gjN ∈ GjN/N and let gj−1N ∈ Gj−1N/N. Then (gjN)(gj−1N)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj. Gj−1N/N is normal in GjN/N: Gj−1N/N is a subgroup of GjN/N. Let gjN ∈ GjN/N and let gj−1N ∈ Gj−1N/N. Then (gjN)(gj−1N) = Ngjgj−1N

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj. Gj−1N/N is normal in GjN/N: Gj−1N/N is a subgroup of GjN/N. Let gjN ∈ GjN/N and let gj−1N ∈ Gj−1N/N. Then (gjN)(gj−1N) = Ngjgj−1N = Ng′

j−1gjN

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj. Gj−1N/N is normal in GjN/N: Gj−1N/N is a subgroup of GjN/N. Let gjN ∈ GjN/N and let gj−1N ∈ Gj−1N/N. Then (gjN)(gj−1N) = Ngjgj−1N = Ng′

j−1gjN =

• g′

j−1N

• (gjN)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj. Gj−1N/N is normal in GjN/N: Gj−1N/N is a subgroup of GjN/N. Let gjN ∈ GjN/N and let gj−1N ∈ Gj−1N/N. Then (gjN)(gj−1N) = Ngjgj−1N = Ng′

j−1gjN =

• g′

j−1N

• (gjN),

which proves that (gjN)(Gj−1N/N) ⊆ (Gj−1N/N)(gjN).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj. Gj−1N/N is normal in GjN/N: Gj−1N/N is a subgroup of GjN/N. Let gjN ∈ GjN/N and let gj−1N ∈ Gj−1N/N. Then (gjN)(gj−1N) = Ngjgj−1N = Ng′

j−1gjN =

• g′

j−1N

• (gjN),

which proves that (gjN)(Gj−1N/N) ⊆ (Gj−1N/N)(gjN). Thus for all x ∈ GjN/N we have x(Gj−1N/N) ⊆ (Gj−1N/N)x

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj. Gj−1N/N is normal in GjN/N: Gj−1N/N is a subgroup of GjN/N. Let gjN ∈ GjN/N and let gj−1N ∈ Gj−1N/N. Then (gjN)(gj−1N) = Ngjgj−1N = Ng′

j−1gjN =

• g′

j−1N

• (gjN),

which proves that (gjN)(Gj−1N/N) ⊆ (Gj−1N/N)(gjN). Thus for all x ∈ GjN/N we have x(Gj−1N/N) ⊆ (Gj−1N/N)x, that is, x(Gj−1N/N)x−1 ⊆ (Gj−1N/N)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Proof. Let {e} = G0 ⊳G1 ⊳···⊳Gm = G be a chain of

subgroups so that Gj/Gj−1 is commutative for j = 1,...,m. Each GjN is a subgroup of G: e ∈ GjN = /

• 0. Let x,y ∈ GjN.

Then there are gx,gy ∈ Gj and nx,ny ∈ N so that x = gxnx and y = gyny. Now xy = (gxnx)(gyny) = gx(nxgy)ny = gx(gyn)ny = (gxgy)(nny) ∈ GjN and x−1 = n−1

x g−1 x

= g−1

x ˜

n ∈ GjN. N ⊳G, so N ⊳GjN. If xN ∈ GjN/N, then there are a gj ∈ Gj and an n ∈ N so that xN = gjnN = gjN. So each element of GjN/N is of the form gjN for some gj ∈ Gj. Gj−1N/N is normal in GjN/N: Gj−1N/N is a subgroup of GjN/N. Let gjN ∈ GjN/N and let gj−1N ∈ Gj−1N/N. Then (gjN)(gj−1N) = Ngjgj−1N = Ng′

j−1gjN =

• g′

j−1N

• (gjN),

which proves that (gjN)(Gj−1N/N) ⊆ (Gj−1N/N)(gjN). Thus for all x ∈ GjN/N we have x(Gj−1N/N) ⊆ (Gj−1N/N)x, that is, x(Gj−1N/N)x−1 ⊆ (Gj−1N/N) and Gj−1N/N ⊳GjN/N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative:

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N with gj ∈ Gj.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N with gj ∈ Gj. Now let gjGj−1N,hjGj−1N ∈ (GjN/N)/(Gj−1N/N).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N with gj ∈ Gj. Now let gjGj−1N,hjGj−1N ∈ (GjN/N)/(Gj−1N/N). Then (gjGj−1N)(hjGj−1N)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N with gj ∈ Gj. Now let gjGj−1N,hjGj−1N ∈ (GjN/N)/(Gj−1N/N). Then (gjGj−1N)(hjGj−1N) = gjGj−1hjNGj−1N

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N with gj ∈ Gj. Now let gjGj−1N,hjGj−1N ∈ (GjN/N)/(Gj−1N/N). Then (gjGj−1N)(hjGj−1N) = gjGj−1hjNGj−1N = gjGj−1hjGj−1NN

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N with gj ∈ Gj. Now let gjGj−1N,hjGj−1N ∈ (GjN/N)/(Gj−1N/N). Then (gjGj−1N)(hjGj−1N) = gjGj−1hjNGj−1N = gjGj−1hjGj−1NN = hjGj−1gjGj−1NN

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N with gj ∈ Gj. Now let gjGj−1N,hjGj−1N ∈ (GjN/N)/(Gj−1N/N). Then (gjGj−1N)(hjGj−1N) = gjGj−1hjNGj−1N = gjGj−1hjGj−1NN = hjGj−1gjGj−1NN = hjGj−1gjNGj−1N

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N with gj ∈ Gj. Now let gjGj−1N,hjGj−1N ∈ (GjN/N)/(Gj−1N/N). Then (gjGj−1N)(hjGj−1N) = gjGj−1hjNGj−1N = gjGj−1hjGj−1NN = hjGj−1gjGj−1NN = hjGj−1gjNGj−1N = (hjGj−1N)(gjGj−1N)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N with gj ∈ Gj. Now let gjGj−1N,hjGj−1N ∈ (GjN/N)/(Gj−1N/N). Then (gjGj−1N)(hjGj−1N) = gjGj−1hjNGj−1N = gjGj−1hjGj−1NN = hjGj−1gjGj−1NN = hjGj−1gjNGj−1N = (hjGj−1N)(gjGj−1N), which means that (GjN/N)/(Gj−1N/N) is commutative.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N with gj ∈ Gj. Now let gjGj−1N,hjGj−1N ∈ (GjN/N)/(Gj−1N/N). Then (gjGj−1N)(hjGj−1N) = gjGj−1hjNGj−1N = gjGj−1hjGj−1NN = hjGj−1gjGj−1NN = hjGj−1gjNGj−1N = (hjGj−1N)(gjGj−1N), which means that (GjN/N)/(Gj−1N/N) is commutative. Therefore, {eN} = G0N/N ⊳G1N/N ⊳···⊳GmN/N = G/N is a chain of subgroups as in the definition of solvability.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). (GjN/N)/(Gj−1N/N) is commutative: The elements of (GjN/N)/(Gj−1N/N) are of the form gjNGj−1N with gj ∈ Gj. Hence each element of (GjN/N)/(Gj−1N/N) is of the form gjNGj−1N = gjGj−1NN = gjGj−1N with gj ∈ Gj. Now let gjGj−1N,hjGj−1N ∈ (GjN/N)/(Gj−1N/N). Then (gjGj−1N)(hjGj−1N) = gjGj−1hjNGj−1N = gjGj−1hjGj−1NN = hjGj−1gjGj−1NN = hjGj−1gjNGj−1N = (hjGj−1N)(gjGj−1N), which means that (GjN/N)/(Gj−1N/N) is commutative. Therefore, {eN} = G0N/N ⊳G1N/N ⊳···⊳GmN/N = G/N is a chain of subgroups as in the definition of solvability.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq that has a root tower over F.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq that has a root tower over F. Proof (q ≤ 2).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq that has a root tower over F. Proof (q ≤ 2). Because q = [Fq : F]

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq that has a root tower over F. Proof (q ≤ 2). Because q = [Fq : F], the case q = 1 is trivial

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq that has a root tower over F. Proof (q ≤ 2). Because q = [Fq : F], the case q = 1 is trivial (F is the splitting field of x−1).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq that has a root tower over F. Proof (q ≤ 2). Because q = [Fq : F], the case q = 1 is trivial (F is the splitting field of x−1). In case q = 2, F2 = F(w2) for some w2 ∈ F2 with wn1

2 ∈ F.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq that has a root tower over F. Proof (q ≤ 2). Because q = [Fq : F], the case q = 1 is trivial (F is the splitting field of x−1). In case q = 2, F2 = F(w2) for some w2 ∈ F2 with wn1

2 ∈ F. If w2

is a primitive nst

1 root of unity, let E := F2.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq that has a root tower over F. Proof (q ≤ 2). Because q = [Fq : F], the case q = 1 is trivial (F is the splitting field of x−1). In case q = 2, F2 = F(w2) for some w2 ∈ F2 with wn1

2 ∈ F. If w2

is a primitive nst

1 root of unity, let E := F2. Otherwise, let θ be a

primitive nst

1 root of unity and let E := F2(θ).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq that has a root tower over F. Proof (q ≤ 2). Because q = [Fq : F], the case q = 1 is trivial (F is the splitting field of x−1). In case q = 2, F2 = F(w2) for some w2 ∈ F2 with wn1

2 ∈ F. If w2

is a primitive nst

1 root of unity, let E := F2. Otherwise, let θ be a

primitive nst

1 root of unity and let E := F2(θ). In either case, E

is a normal extension of F2

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq that has a root tower over F. Proof (q ≤ 2). Because q = [Fq : F], the case q = 1 is trivial (F is the splitting field of x−1). In case q = 2, F2 = F(w2) for some w2 ∈ F2 with wn1

2 ∈ F. If w2

is a primitive nst

1 root of unity, let E := F2. Otherwise, let θ be a

primitive nst

1 root of unity and let E := F2(θ). In either case, E

is a normal extension of F2 (because E is the splitting field of xn1 −1)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

F = F1 ⊆ F2 ⊆ ··· ⊆ Fq be a root tower for Fq over F. Then there is normal extension E of Fq that has a root tower over F. Proof (q ≤ 2). Because q = [Fq : F], the case q = 1 is trivial (F is the splitting field of x−1). In case q = 2, F2 = F(w2) for some w2 ∈ F2 with wn1

2 ∈ F. If w2

is a primitive nst

1 root of unity, let E := F2. Otherwise, let θ be a

primitive nst

1 root of unity and let E := F2(θ). In either case, E

is a normal extension of F2 (because E is the splitting field of xn1 −1) that has the root tower F ⊆ F2 ⊆ E over F.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction). Let F′

q−1 be a normal extension of

Fq−1 that has a root tower over F.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction). Let F′

q−1 be a normal extension of

Fq−1 that has a root tower over F. By assumption, Fq = Fq−1(wq) for some wq with w

nq−1 q

∈ Fq−1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction). Let F′

q−1 be a normal extension of

Fq−1 that has a root tower over F. By assumption, Fq = Fq−1(wq) for some wq with w

nq−1 q

∈ Fq−1. Let f(x) :=

∏

σ∈G(F′

q−1/F)

• xnq−1 −σ
• w

nq−1 q

• .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction). Let F′

q−1 be a normal extension of

Fq−1 that has a root tower over F. By assumption, Fq = Fq−1(wq) for some wq with w

nq−1 q

∈ Fq−1. Let f(x) :=

∏

σ∈G(F′

q−1/F)

• xnq−1 −σ
• w

nq−1 q

• . For all

ϕ ∈ G(F′

q−1/F) we have

˜ ϕ(f)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction). Let F′

q−1 be a normal extension of

Fq−1 that has a root tower over F. By assumption, Fq = Fq−1(wq) for some wq with w

nq−1 q

∈ Fq−1. Let f(x) :=

∏

σ∈G(F′

q−1/F)

• xnq−1 −σ
• w

nq−1 q

• . For all

ϕ ∈ G(F′

q−1/F) we have

˜ ϕ(f) =

∏

σ∈G(F′

q−1/F)

• xnq−1 −ϕσ
• w

nq−1 q

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction). Let F′

q−1 be a normal extension of

Fq−1 that has a root tower over F. By assumption, Fq = Fq−1(wq) for some wq with w

nq−1 q

∈ Fq−1. Let f(x) :=

∏

σ∈G(F′

q−1/F)

• xnq−1 −σ
• w

nq−1 q

• . For all

ϕ ∈ G(F′

q−1/F) we have

˜ ϕ(f) =

∏

σ∈G(F′

q−1/F)

• xnq−1 −ϕσ
• w

nq−1 q

• =

∏

γ∈G(F′

q−1/F)

• xnq−1 −γ
• w

nq−1 q

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction). Let F′

q−1 be a normal extension of

Fq−1 that has a root tower over F. By assumption, Fq = Fq−1(wq) for some wq with w

nq−1 q

∈ Fq−1. Let f(x) :=

∏

σ∈G(F′

q−1/F)

• xnq−1 −σ
• w

nq−1 q

• . For all

ϕ ∈ G(F′

q−1/F) we have

˜ ϕ(f) =

∏

σ∈G(F′

q−1/F)

• xnq−1 −ϕσ
• w

nq−1 q

• =

∏

γ∈G(F′

q−1/F)

• xnq−1 −γ
• w

nq−1 q

• = f

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction). Let F′

q−1 be a normal extension of

Fq−1 that has a root tower over F. By assumption, Fq = Fq−1(wq) for some wq with w

nq−1 q

∈ Fq−1. Let f(x) :=

∏

σ∈G(F′

q−1/F)

• xnq−1 −σ
• w

nq−1 q

• . For all

ϕ ∈ G(F′

q−1/F) we have

˜ ϕ(f) =

∏

σ∈G(F′

q−1/F)

• xnq−1 −ϕσ
• w

nq−1 q

• =

∏

γ∈G(F′

q−1/F)

• xnq−1 −γ
• w

nq−1 q

• = f,

so the coefficients of f are in the fixed field F of G(F′

q−1/F).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction). Let F′

q−1 be a normal extension of

Fq−1 that has a root tower over F. By assumption, Fq = Fq−1(wq) for some wq with w

nq−1 q

∈ Fq−1. Let f(x) :=

∏

σ∈G(F′

q−1/F)

• xnq−1 −σ
• w

nq−1 q

• . For all

ϕ ∈ G(F′

q−1/F) we have

˜ ϕ(f) =

∏

σ∈G(F′

q−1/F)

• xnq−1 −ϕσ
• w

nq−1 q

• =

∏

γ∈G(F′

q−1/F)

• xnq−1 −γ
• w

nq−1 q

• = f,

so the coefficients of f are in the fixed field F of G(F′

q−1/F).

Hence f ∈ F[x].

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x].

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x]. The polynomial fg ∈ F[x] splits in E

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x]. The polynomial fg ∈ F[x] splits in E and any field in which fg splits must contain F′

q−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x]. The polynomial fg ∈ F[x] splits in E and any field in which fg splits must contain F′

q−1 as well as the elements that must be adjoined to it to

• btain E.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x]. The polynomial fg ∈ F[x] splits in E and any field in which fg splits must contain F′

q−1 as well as the elements that must be adjoined to it to

• btain E. Hence E is the splitting field of fg

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x]. The polynomial fg ∈ F[x] splits in E and any field in which fg splits must contain F′

q−1 as well as the elements that must be adjoined to it to

• btain E. Hence E is the splitting field of fg and thus E is a

normal extension of F.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x]. The polynomial fg ∈ F[x] splits in E and any field in which fg splits must contain F′

q−1 as well as the elements that must be adjoined to it to

• btain E. Hence E is the splitting field of fg and thus E is a

normal extension of F. By definition of f, E is obtained from F′

q−1 by adjoining nst q−1

roots.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x]. The polynomial fg ∈ F[x] splits in E and any field in which fg splits must contain F′

q−1 as well as the elements that must be adjoined to it to

• btain E. Hence E is the splitting field of fg and thus E is a

normal extension of F. By definition of f, E is obtained from F′

q−1 by adjoining nst q−1

• roots. Hence E has a root tower over F′

q−1.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x]. The polynomial fg ∈ F[x] splits in E and any field in which fg splits must contain F′

q−1 as well as the elements that must be adjoined to it to

• btain E. Hence E is the splitting field of fg and thus E is a

normal extension of F. By definition of f, E is obtained from F′

q−1 by adjoining nst q−1

• roots. Hence E has a root tower over F′

q−1. Appending this root

tower to the root tower for F′

q−1 over F produces a root tower

for E over F.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x]. The polynomial fg ∈ F[x] splits in E and any field in which fg splits must contain F′

q−1 as well as the elements that must be adjoined to it to

• btain E. Hence E is the splitting field of fg and thus E is a

normal extension of F. By definition of f, E is obtained from F′

q−1 by adjoining nst q−1

• roots. Hence E has a root tower over F′

q−1. Appending this root

tower to the root tower for F′

q−1 over F produces a root tower

for E over F. Finally, wq is a root of f.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x]. The polynomial fg ∈ F[x] splits in E and any field in which fg splits must contain F′

q−1 as well as the elements that must be adjoined to it to

• btain E. Hence E is the splitting field of fg and thus E is a

normal extension of F. By definition of f, E is obtained from F′

q−1 by adjoining nst q−1

• roots. Hence E has a root tower over F′

q−1. Appending this root

tower to the root tower for F′

q−1 over F produces a root tower

for E over F. Finally, wq is a root of f. Hence E must contain the field Fq = Fq−1(wq).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (q > 2, induction, concl.). Let E be the splitting field of f over F′

q−1. Because F′ q−1 is a normal extension of F, it is the

splitting field of a polynomial g ∈ F[x]. The polynomial fg ∈ F[x] splits in E and any field in which fg splits must contain F′

q−1 as well as the elements that must be adjoined to it to

• btain E. Hence E is the splitting field of fg and thus E is a

normal extension of F. By definition of f, E is obtained from F′

q−1 by adjoining nst q−1

• roots. Hence E has a root tower over F′

q−1. Appending this root

tower to the root tower for F′

q−1 over F produces a root tower

for E over F. Finally, wq is a root of f. Hence E must contain the field Fq = Fq−1(wq).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

• Proof. Let F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E be a root tower for E
• ver F

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

• Proof. Let F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E be a root tower for E
• ver F, let n := n1 ···nq−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

• Proof. Let F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E be a root tower for E
• ver F, let n := n1 ···nq−1 and let θ be a primitive nth root of

unity.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

• Proof. Let F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E be a root tower for E
• ver F, let n := n1 ···nq−1 and let θ be a primitive nth root of
• unity. For all j ∈ {2,...,q} we have Fj(θ) = Fj−1(θ)(wj)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

• Proof. Let F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E be a root tower for E
• ver F, let n := n1 ···nq−1 and let θ be a primitive nth root of
• unity. For all j ∈ {2,...,q} we have Fj(θ) = Fj−1(θ)(wj) and

θ

n1···nq−1 nj−1

∈ Fj−1(θ) is a primitive nst

j−1 root of unity.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

• Proof. Let F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E be a root tower for E
• ver F, let n := n1 ···nq−1 and let θ be a primitive nth root of
• unity. For all j ∈ {2,...,q} we have Fj(θ) = Fj−1(θ)(wj) and

θ

n1···nq−1 nj−1

∈ Fj−1(θ) is a primitive nst

j−1 root of unity. Thus Fj(θ)

is a normal extension of Fj−1(θ)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

• Proof. Let F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E be a root tower for E
• ver F, let n := n1 ···nq−1 and let θ be a primitive nth root of
• unity. For all j ∈ {2,...,q} we have Fj(θ) = Fj−1(θ)(wj) and

θ

n1···nq−1 nj−1

∈ Fj−1(θ) is a primitive nst

j−1 root of unity. Thus Fj(θ)

is a normal extension of Fj−1(θ) and it has a commutative Galois group G

• Fj(θ)/Fj−1(θ)
• .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

• Proof. Let F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E be a root tower for E
• ver F, let n := n1 ···nq−1 and let θ be a primitive nth root of
• unity. For all j ∈ {2,...,q} we have Fj(θ) = Fj−1(θ)(wj) and

θ

n1···nq−1 nj−1

∈ Fj−1(θ) is a primitive nst

j−1 root of unity. Thus Fj(θ)

is a normal extension of Fj−1(θ) and it has a commutative Galois group G

• Fj(θ)/Fj−1(θ)
• .

E is the splitting field of a polynomial g ∈ F[x].

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

• Proof. Let F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E be a root tower for E
• ver F, let n := n1 ···nq−1 and let θ be a primitive nth root of
• unity. For all j ∈ {2,...,q} we have Fj(θ) = Fj−1(θ)(wj) and

θ

n1···nq−1 nj−1

∈ Fj−1(θ) is a primitive nst

j−1 root of unity. Thus Fj(θ)

is a normal extension of Fj−1(θ) and it has a commutative Galois group G

• Fj(θ)/Fj−1(θ)
• .

E is the splitting field of a polynomial g ∈ F[x]. Then E(θ) is the splitting field of g(x)(xn −1).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

• Proof. Let F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E be a root tower for E
• ver F, let n := n1 ···nq−1 and let θ be a primitive nth root of
• unity. For all j ∈ {2,...,q} we have Fj(θ) = Fj−1(θ)(wj) and

θ

n1···nq−1 nj−1

∈ Fj−1(θ) is a primitive nst

j−1 root of unity. Thus Fj(θ)

is a normal extension of Fj−1(θ) and it has a commutative Galois group G

• Fj(θ)/Fj−1(θ)
• .

E is the splitting field of a polynomial g ∈ F[x]. Then E(θ) is the splitting field of g(x)(xn −1). Hence E(θ) is a normal extension of F

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let F be a field of characteristic 0 and let E be a

normal extension with a root tower over F. Then the Galois group G(E/F) is solvable.

• Proof. Let F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E be a root tower for E
• ver F, let n := n1 ···nq−1 and let θ be a primitive nth root of
• unity. For all j ∈ {2,...,q} we have Fj(θ) = Fj−1(θ)(wj) and

θ

n1···nq−1 nj−1

∈ Fj−1(θ) is a primitive nst

j−1 root of unity. Thus Fj(θ)

is a normal extension of Fj−1(θ) and it has a commutative Galois group G

• Fj(θ)/Fj−1(θ)
• .

E is the splitting field of a polynomial g ∈ F[x]. Then E(θ) is the splitting field of g(x)(xn −1). Hence E(θ) is a normal extension of F and of each Fj(θ).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Moreover, F(θ) is a normal extension of F

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Moreover, F(θ) is a normal extension of F (it’s the splitting field of xn −θ n

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Moreover, F(θ) is a normal extension of F (it’s the splitting field of xn −θ n = xn −1)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Moreover, F(θ) is a normal extension of F (it’s the splitting field of xn −θ n = xn −1) and G

• E(θ)/F
• /G
• E(θ)/F(θ)
• is

isomorphic to G

• F(θ)/F
• .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Moreover, F(θ) is a normal extension of F (it’s the splitting field of xn −θ n = xn −1) and G

• E(θ)/F
• /G
• E(θ)/F(θ)
• is

isomorphic to G

• F(θ)/F
• . This group is commutative

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Moreover, F(θ) is a normal extension of F (it’s the splitting field of xn −θ n = xn −1) and G

• E(θ)/F
• /G
• E(θ)/F(θ)
• is

isomorphic to G

• F(θ)/F
• . This group is commutative (good

exercise, similar to the proof of this presentation’s first theorem).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Moreover, F(θ) is a normal extension of F (it’s the splitting field of xn −θ n = xn −1) and G

• E(θ)/F
• /G
• E(θ)/F(θ)
• is

isomorphic to G

• F(θ)/F
• . This group is commutative (good

exercise, similar to the proof of this presentation’s first theorem). Hence G

• E(θ)/F
• is solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Moreover, F(θ) is a normal extension of F (it’s the splitting field of xn −θ n = xn −1) and G

• E(θ)/F
• /G
• E(θ)/F(θ)
• is

isomorphic to G

• F(θ)/F
• . This group is commutative (good

exercise, similar to the proof of this presentation’s first theorem). Hence G

• E(θ)/F
• is solvable.

By hypothesis, E is a normal extension of F

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Moreover, F(θ) is a normal extension of F (it’s the splitting field of xn −θ n = xn −1) and G

• E(θ)/F
• /G
• E(θ)/F(θ)
• is

isomorphic to G

• F(θ)/F
• . This group is commutative (good

exercise, similar to the proof of this presentation’s first theorem). Hence G

• E(θ)/F
• is solvable.

By hypothesis, E is a normal extension of F, and by the Fundamental Theorem of Galois Theory G(E/F) is isomorphic to G

• E(θ)/F
• /G
• E(θ)/E)
• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Moreover, F(θ) is a normal extension of F (it’s the splitting field of xn −θ n = xn −1) and G

• E(θ)/F
• /G
• E(θ)/F(θ)
• is

isomorphic to G

• F(θ)/F
• . This group is commutative (good

exercise, similar to the proof of this presentation’s first theorem). Hence G

• E(θ)/F
• is solvable.

By hypothesis, E is a normal extension of F, and by the Fundamental Theorem of Galois Theory G(E/F) is isomorphic to G

• E(θ)/F
• /G
• E(θ)/E)
• , which is solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Proof (concl.). In the root tower F ⊆ F(θ) = F1(θ) ⊆ F2(θ) ⊆ ··· ⊆ Fq(θ) = E(θ), for each j ∈ {2,...,q} the group G

• E(θ)/Fj(θ)
• is a normal subgroup
• f G
• E(θ)/Fj−1(θ)
• and G
• E(θ)/Fj−1(θ)
• /G
• E(θ)/Fj(θ)
• is isomorphic to G
• Fj(θ)/Fj−1(θ)
• , which is commutative.

Moreover, F(θ) is a normal extension of F (it’s the splitting field of xn −θ n = xn −1) and G

• E(θ)/F
• /G
• E(θ)/F(θ)
• is

isomorphic to G

• F(θ)/F
• . This group is commutative (good

exercise, similar to the proof of this presentation’s first theorem). Hence G

• E(θ)/F
• is solvable.

By hypothesis, E is a normal extension of F, and by the Fundamental Theorem of Galois Theory G(E/F) is isomorphic to G

• E(θ)/F
• /G
• E(θ)/E)
• , which is solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

p ∈ F[x] be so that p(x) = 0 is solvable by radicals.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

p ∈ F[x] be so that p(x) = 0 is solvable by radicals. Then the Galois group of p is solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

p ∈ F[x] be so that p(x) = 0 is solvable by radicals. Then the Galois group of p is solvable. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

p ∈ F[x] be so that p(x) = 0 is solvable by radicals. Then the Galois group of p is solvable.

• Proof. Let S be the splitting field of p over F.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

p ∈ F[x] be so that p(x) = 0 is solvable by radicals. Then the Galois group of p is solvable.

• Proof. Let S be the splitting field of p over F. Because there is a

root tower over F whose top field contains all the roots of p, there is a normal extension E of S that has a root tower F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

p ∈ F[x] be so that p(x) = 0 is solvable by radicals. Then the Galois group of p is solvable.

• Proof. Let S be the splitting field of p over F. Because there is a

root tower over F whose top field contains all the roots of p, there is a normal extension E of S that has a root tower F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E. Moreover, G(E/F) is solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

p ∈ F[x] be so that p(x) = 0 is solvable by radicals. Then the Galois group of p is solvable.

• Proof. Let S be the splitting field of p over F. Because there is a

root tower over F whose top field contains all the roots of p, there is a normal extension E of S that has a root tower F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E. Moreover, G(E/F) is solvable. By the Fundamental Theorem of Galois Theory, G(S/F) is isomorphic to G(E/F)/G(E/S)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

p ∈ F[x] be so that p(x) = 0 is solvable by radicals. Then the Galois group of p is solvable.

• Proof. Let S be the splitting field of p over F. Because there is a

root tower over F whose top field contains all the roots of p, there is a normal extension E of S that has a root tower F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E. Moreover, G(E/F) is solvable. By the Fundamental Theorem of Galois Theory, G(S/F) is isomorphic to G(E/F)/G(E/S), which is solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals

logo1 Commutative Galois Groups Solvable Groups Better Root Towers Solvable Galois Groups

• Theorem. Let (F,+,·) be a field of characteristic 0 and let

p ∈ F[x] be so that p(x) = 0 is solvable by radicals. Then the Galois group of p is solvable.

• Proof. Let S be the splitting field of p over F. Because there is a

root tower over F whose top field contains all the roots of p, there is a normal extension E of S that has a root tower F = F1 ⊆ F2 ⊆ ··· ⊆ Fq = E. Moreover, G(E/F) is solvable. By the Fundamental Theorem of Galois Theory, G(S/F) is isomorphic to G(E/F)/G(E/S), which is solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Consequences of Solvability by Radicals