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logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Abel’s Theorem

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Introduction

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Introduction

• 1. If p(x) = 0 is solvable by radicals, then the Galois group of

p must be solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Introduction

• 1. If p(x) = 0 is solvable by radicals, then the Galois group of

p must be solvable.

• 2. There is an irreducible polynomial with Galois group

isomorphic to S5.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Introduction

• 1. If p(x) = 0 is solvable by radicals, then the Galois group of

p must be solvable.

• 2. There is an irreducible polynomial with Galois group

isomorphic to S5.

• 3. But S5 does not have many normal subgroups, so we have

a chance to determine if it is solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Lemma.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

• Proof. As an intersection of two subgroups

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

• Proof. As an intersection of two subgroups, H ∩N is a

subgroup, too.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

• Proof. As an intersection of two subgroups, H ∩N is a

subgroup, too. Let x ∈ N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

• Proof. As an intersection of two subgroups, H ∩N is a

subgroup, too. Let x ∈ N. Because H is normal in G, we have xHx−1 = H.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

• Proof. As an intersection of two subgroups, H ∩N is a

subgroup, too. Let x ∈ N. Because H is normal in G, we have xHx−1 = H. So xhx−1 ∈ H for all h ∈ H.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

• Proof. As an intersection of two subgroups, H ∩N is a

subgroup, too. Let x ∈ N. Because H is normal in G, we have xHx−1 = H. So xhx−1 ∈ H for all h ∈ H. Let y ∈ H ∩N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

• Proof. As an intersection of two subgroups, H ∩N is a

subgroup, too. Let x ∈ N. Because H is normal in G, we have xHx−1 = H. So xhx−1 ∈ H for all h ∈ H. Let y ∈ H ∩N. By what we just proved, xyx−1 ∈ H.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

• Proof. As an intersection of two subgroups, H ∩N is a

subgroup, too. Let x ∈ N. Because H is normal in G, we have xHx−1 = H. So xhx−1 ∈ H for all h ∈ H. Let y ∈ H ∩N. By what we just proved, xyx−1 ∈ H. Moreover, because N is a group, we have xyx−1 ∈ N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

• Proof. As an intersection of two subgroups, H ∩N is a

subgroup, too. Let x ∈ N. Because H is normal in G, we have xHx−1 = H. So xhx−1 ∈ H for all h ∈ H. Let y ∈ H ∩N. By what we just proved, xyx−1 ∈ H. Moreover, because N is a group, we have xyx−1 ∈ N. Thus xyx−1 ∈ H ∩N for all y ∈ H ∩N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

• Proof. As an intersection of two subgroups, H ∩N is a

subgroup, too. Let x ∈ N. Because H is normal in G, we have xHx−1 = H. So xhx−1 ∈ H for all h ∈ H. Let y ∈ H ∩N. By what we just proved, xyx−1 ∈ H. Moreover, because N is a group, we have xyx−1 ∈ N. Thus xyx−1 ∈ H ∩N for all y ∈ H ∩N. Because x ∈ N was arbitrary, we conclude that H ∩N is a normal subgroup of N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Lemma. Let G be a group and let H and N be normal

subgroups of G. Then H ∩N is a normal subgroup of N.

• Proof. As an intersection of two subgroups, H ∩N is a

subgroup, too. Let x ∈ N. Because H is normal in G, we have xHx−1 = H. So xhx−1 ∈ H for all h ∈ H. Let y ∈ H ∩N. By what we just proved, xyx−1 ∈ H. Moreover, because N is a group, we have xyx−1 ∈ N. Thus xyx−1 ∈ H ∩N for all y ∈ H ∩N. Because x ∈ N was arbitrary, we conclude that H ∩N is a normal subgroup of N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}. Then N ∩An is a normal subgroup of An.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}. Then N ∩An is a normal subgroup of An. Because An is simple, N ∩An ∈

• {id},An
• .

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}. Then N ∩An is a normal subgroup of An. Because An is simple, N ∩An ∈

• {id},An
• .

Suppose for a contradiction that N ∩An = {id}.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}. Then N ∩An is a normal subgroup of An. Because An is simple, N ∩An ∈

• {id},An
• .

Suppose for a contradiction that N ∩An = {id}. Then all nontrivial permutations in N are odd.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}. Then N ∩An is a normal subgroup of An. Because An is simple, N ∩An ∈

• {id},An
• .

Suppose for a contradiction that N ∩An = {id}. Then all nontrivial permutations in N are odd. Let σ ∈ N \{id}.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}. Then N ∩An is a normal subgroup of An. Because An is simple, N ∩An ∈

• {id},An
• .

Suppose for a contradiction that N ∩An = {id}. Then all nontrivial permutations in N are odd. Let σ ∈ N \{id}. Then σ2 ∈ N must be even.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}. Then N ∩An is a normal subgroup of An. Because An is simple, N ∩An ∈

• {id},An
• .

Suppose for a contradiction that N ∩An = {id}. Then all nontrivial permutations in N are odd. Let σ ∈ N \{id}. Then σ2 ∈ N must be even. Hence σ2 = id.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}. Then N ∩An is a normal subgroup of An. Because An is simple, N ∩An ∈

• {id},An
• .

Suppose for a contradiction that N ∩An = {id}. Then all nontrivial permutations in N are odd. Let σ ∈ N \{id}. Then σ2 ∈ N must be even. Hence σ2 = id. Any odd permutation is a composition of disjoint cycles.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}. Then N ∩An is a normal subgroup of An. Because An is simple, N ∩An ∈

• {id},An
• .

Suppose for a contradiction that N ∩An = {id}. Then all nontrivial permutations in N are odd. Let σ ∈ N \{id}. Then σ2 ∈ N must be even. Hence σ2 = id. Any odd permutation is a composition of disjoint cycles. So the square of each cycle in the representation of σ must be the identity.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}. Then N ∩An is a normal subgroup of An. Because An is simple, N ∩An ∈

• {id},An
• .

Suppose for a contradiction that N ∩An = {id}. Then all nontrivial permutations in N are odd. Let σ ∈ N \{id}. Then σ2 ∈ N must be even. Hence σ2 = id. Any odd permutation is a composition of disjoint cycles. So the square of each cycle in the representation of σ must be the identity. But the only cycles whose square is the identity are transpositions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. For every prime number n ≥ 5, the group Sn is not

solvable.

• Proof. Let n ≥ 5 be a prime number. We first prove that {id},

An and Sn are the only normal subgroups of Sn. Let N ⊳Sn be a normal subgroup of Sn with N = {id}. Then N ∩An is a normal subgroup of An. Because An is simple, N ∩An ∈

• {id},An
• .

Suppose for a contradiction that N ∩An = {id}. Then all nontrivial permutations in N are odd. Let σ ∈ N \{id}. Then σ2 ∈ N must be even. Hence σ2 = id. Any odd permutation is a composition of disjoint cycles. So the square of each cycle in the representation of σ must be the identity. But the only cycles whose square is the identity are transpositions. Therefore all nontrivial elements of N are products of an odd number of disjoint transpositions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be. Thus τ is a product of an odd number (greater than 1) of disjoint transpositions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be. Thus τ is a product of an odd number (greater than 1) of disjoint

• transpositions. Without loss of generality assume that

τ = (12)(34)δ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be. Thus τ is a product of an odd number (greater than 1) of disjoint

• transpositions. Without loss of generality assume that

τ = (12)(34)δ, where δ is a product of an odd number of disjoint transpositions that do not contain any of the numbers 1,2,3,4.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be. Thus τ is a product of an odd number (greater than 1) of disjoint

• transpositions. Without loss of generality assume that

τ = (12)(34)δ, where δ is a product of an odd number of disjoint transpositions that do not contain any of the numbers 1,2,3,4. Then, because N is normal, (13)(24)δ

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be. Thus τ is a product of an odd number (greater than 1) of disjoint

• transpositions. Without loss of generality assume that

τ = (12)(34)δ, where δ is a product of an odd number of disjoint transpositions that do not contain any of the numbers 1,2,3,4. Then, because N is normal, (13)(24)δ = (23)(12)(34)δ(23)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be. Thus τ is a product of an odd number (greater than 1) of disjoint

• transpositions. Without loss of generality assume that

τ = (12)(34)δ, where δ is a product of an odd number of disjoint transpositions that do not contain any of the numbers 1,2,3,4. Then, because N is normal, (13)(24)δ = (23)(12)(34)δ(23) ∈ N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be. Thus τ is a product of an odd number (greater than 1) of disjoint

• transpositions. Without loss of generality assume that

τ = (12)(34)δ, where δ is a product of an odd number of disjoint transpositions that do not contain any of the numbers 1,2,3,4. Then, because N is normal, (13)(24)δ = (23)(12)(34)δ(23) ∈ N. Because N is a subgroup, (14)(23)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be. Thus τ is a product of an odd number (greater than 1) of disjoint

• transpositions. Without loss of generality assume that

τ = (12)(34)δ, where δ is a product of an odd number of disjoint transpositions that do not contain any of the numbers 1,2,3,4. Then, because N is normal, (13)(24)δ = (23)(12)(34)δ(23) ∈ N. Because N is a subgroup, (14)(23) = (12)(34)(24)(13)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be. Thus τ is a product of an odd number (greater than 1) of disjoint

• transpositions. Without loss of generality assume that

τ = (12)(34)δ, where δ is a product of an odd number of disjoint transpositions that do not contain any of the numbers 1,2,3,4. Then, because N is normal, (13)(24)δ = (23)(12)(34)δ(23) ∈ N. Because N is a subgroup, (14)(23) = (12)(34)(24)(13) = (12)(34)δδ −1(24)(13)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be. Thus τ is a product of an odd number (greater than 1) of disjoint

• transpositions. Without loss of generality assume that

τ = (12)(34)δ, where δ is a product of an odd number of disjoint transpositions that do not contain any of the numbers 1,2,3,4. Then, because N is normal, (13)(24)δ = (23)(12)(34)δ(23) ∈ N. Because N is a subgroup, (14)(23) = (12)(34)(24)(13) = (12)(34)δδ −1(24)(13) ∈ N.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (cont.). Without loss of generality, let τ ∈ N be so that (12) is one of the transpositions that make up τ. Then τ = (12): Indeed, otherwise the normality of N implies that (13) = (32)(12)(23) ∈ N, because N is a subgroup, the even permutation (132) = (13)(23) is in N, which cannot be. Thus τ is a product of an odd number (greater than 1) of disjoint

• transpositions. Without loss of generality assume that

τ = (12)(34)δ, where δ is a product of an odd number of disjoint transpositions that do not contain any of the numbers 1,2,3,4. Then, because N is normal, (13)(24)δ = (23)(12)(34)δ(23) ∈ N. Because N is a subgroup, (14)(23) = (12)(34)(24)(13) = (12)(34)δδ −1(24)(13) ∈ N. But (14)(23) is an even permutation, which is a contradiction to N ∩An = {id}.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.).

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd permutation.

Bernd Schr¨

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Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition. But N also contains all n-cycles, because, for n odd, n-cycles are even.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition. But N also contains all n-cycles, because, for n odd, n-cycles are even. Thus, if N = An, then N = Sn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition. But N also contains all n-cycles, because, for n odd, n-cycles are even. Thus, if N = An, then N = Sn. We have shown that any normal subgroup N of Sn is one of {id}, An and Sn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition. But N also contains all n-cycles, because, for n odd, n-cycles are even. Thus, if N = An, then N = Sn. We have shown that any normal subgroup N of Sn is one of {id}, An and Sn. But neither Sn/{id} ∼ = Sn

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition. But N also contains all n-cycles, because, for n odd, n-cycles are even. Thus, if N = An, then N = Sn. We have shown that any normal subgroup N of Sn is one of {id}, An and Sn. But neither Sn/{id} ∼ = Sn nor An/{id} ∼ = An

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition. But N also contains all n-cycles, because, for n odd, n-cycles are even. Thus, if N = An, then N = Sn. We have shown that any normal subgroup N of Sn is one of {id}, An and Sn. But neither Sn/{id} ∼ = Sn nor An/{id} ∼ = An is commutative.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition. But N also contains all n-cycles, because, for n odd, n-cycles are even. Thus, if N = An, then N = Sn. We have shown that any normal subgroup N of Sn is one of {id}, An and Sn. But neither Sn/{id} ∼ = Sn nor An/{id} ∼ = An is

• commutative. Thus neither of the only two possible chains

{id}⊳Sn

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition. But N also contains all n-cycles, because, for n odd, n-cycles are even. Thus, if N = An, then N = Sn. We have shown that any normal subgroup N of Sn is one of {id}, An and Sn. But neither Sn/{id} ∼ = Sn nor An/{id} ∼ = An is

• commutative. Thus neither of the only two possible chains

{id}⊳Sn and {id}⊳An ⊳Sn

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition. But N also contains all n-cycles, because, for n odd, n-cycles are even. Thus, if N = An, then N = Sn. We have shown that any normal subgroup N of Sn is one of {id}, An and Sn. But neither Sn/{id} ∼ = Sn nor An/{id} ∼ = An is

• commutative. Thus neither of the only two possible chains

{id}⊳Sn and {id}⊳An ⊳Sn of nested normal subgroups from {id} to Sn satisfies the definition of solvability.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition. But N also contains all n-cycles, because, for n odd, n-cycles are even. Thus, if N = An, then N = Sn. We have shown that any normal subgroup N of Sn is one of {id}, An and Sn. But neither Sn/{id} ∼ = Sn nor An/{id} ∼ = An is

• commutative. Thus neither of the only two possible chains

{id}⊳Sn and {id}⊳An ⊳Sn of nested normal subgroups from {id} to Sn satisfies the definition of solvability. Thus Sn is not solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Proof (concl.). Thus N ∩An = {id} and hence N ∩An = An. This means that N is either equal to An, or N contains an odd

• permutation. In case N contains an odd permutation, then,

because N contains An, N must contain a transposition. But N also contains all n-cycles, because, for n odd, n-cycles are even. Thus, if N = An, then N = Sn. We have shown that any normal subgroup N of Sn is one of {id}, An and Sn. But neither Sn/{id} ∼ = Sn nor An/{id} ∼ = An is

• commutative. Thus neither of the only two possible chains

{id}⊳Sn and {id}⊳An ⊳Sn of nested normal subgroups from {id} to Sn satisfies the definition of solvability. Thus Sn is not solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. Abel’s Theorem.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. Abel’s Theorem. There is a fifth order polynomial p

so that p(x) = 0 is not solvable by radicals.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. Abel’s Theorem. There is a fifth order polynomial p

so that p(x) = 0 is not solvable by radicals. In particular, this means that there is no “quintic formula”.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. Abel’s Theorem. There is a fifth order polynomial p

so that p(x) = 0 is not solvable by radicals. In particular, this means that there is no “quintic formula”. Proof.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. Abel’s Theorem. There is a fifth order polynomial p

so that p(x) = 0 is not solvable by radicals. In particular, this means that there is no “quintic formula”.

• Proof. The Galois group of the polynomial

p(x) = x5 −3x4 −3x3 +3x2 +3x+3 from the Eisenstein presentation is isomorphic to S5.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. Abel’s Theorem. There is a fifth order polynomial p

so that p(x) = 0 is not solvable by radicals. In particular, this means that there is no “quintic formula”.

• Proof. The Galois group of the polynomial

p(x) = x5 −3x4 −3x3 +3x2 +3x+3 from the Eisenstein presentation is isomorphic to S5. But S5 is not solvable.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. Abel’s Theorem. There is a fifth order polynomial p

so that p(x) = 0 is not solvable by radicals. In particular, this means that there is no “quintic formula”.

• Proof. The Galois group of the polynomial

p(x) = x5 −3x4 −3x3 +3x2 +3x+3 from the Eisenstein presentation is isomorphic to S5. But S5 is not solvable. If p(x) = 0 was solvable by radicals, we would have a contradiction.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. Abel’s Theorem. There is a fifth order polynomial p

so that p(x) = 0 is not solvable by radicals. In particular, this means that there is no “quintic formula”.

• Proof. The Galois group of the polynomial

p(x) = x5 −3x4 −3x3 +3x2 +3x+3 from the Eisenstein presentation is isomorphic to S5. But S5 is not solvable. If p(x) = 0 was solvable by radicals, we would have a

• contradiction. Hence p(x) = 0 cannot be solvable by radicals.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

• Theorem. Abel’s Theorem. There is a fifth order polynomial p

so that p(x) = 0 is not solvable by radicals. In particular, this means that there is no “quintic formula”.

• Proof. The Galois group of the polynomial

p(x) = x5 −3x4 −3x3 +3x2 +3x+3 from the Eisenstein presentation is isomorphic to S5. But S5 is not solvable. If p(x) = 0 was solvable by radicals, we would have a

• contradiction. Hence p(x) = 0 cannot be solvable by radicals.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

So We Can See the Roots of p(x) = x5 −3x4 −3x3 +3x2 +3x+3 ...

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

So We Can See the Roots of p(x) = x5 −3x4 −3x3 +3x2 +3x+3 ...

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

So We Can See the Roots of p(x) = x5 −3x4 −3x3 +3x2 +3x+3 ...

... but there is no “formula” (can precisely define that) for them.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

So We Can See the Roots of p(x) = x5 −3x4 −3x3 +3x2 +3x+3 ...

... but there is no “formula” (can precisely define that) for them. (Compare with irrationality of √ 2.)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Abel’s Theorem guided focus towards achievable approaches (like approximations).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Abel’s Theorem guided focus towards achievable approaches (like approximations). Galois’ approach to unsolvability by radicals started the path towards better notation and more abstraction, which has significantly advanced mathematics since.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Abel’s Theorem guided focus towards achievable approaches (like approximations). Galois’ approach to unsolvability by radicals started the path towards better notation and more abstraction, which has significantly advanced mathematics since.

• Definition. In honor of Abel, commutative groups are also

called abelian groups.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem

logo1 S5 Is Not Solvable Unsolvability of Quintics by Radicals

Abel’s Theorem guided focus towards achievable approaches (like approximations). Galois’ approach to unsolvability by radicals started the path towards better notation and more abstraction, which has significantly advanced mathematics since.

• Definition. In honor of Abel, commutative groups are also

called abelian groups. In honor of Galois, the work that developed from his proof, which was used to write this proof, is called Galois Theory.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Abel’s Theorem