Review on the basic properties of fields Instructor: Yifan Yang - - PowerPoint PPT Presentation

Review on the basic properties of fields

Instructor: Yifan Yang Spring 2007

Instructor: Yifan Yang Review on the basic properties of fields

Fields

Definition A field F is a commutative ring with unity 1 = 0 such that every nonzero element has a multiplicative inverse, i.e., every nonzero element is a unit. Example

1

Q, R, and C are fields.

2

Zp are fields if and only if p is a prime number.

3

Z, F[x] are not fields.

Instructor: Yifan Yang Review on the basic properties of fields

Fields

Definition A field F is a commutative ring with unity 1 = 0 such that every nonzero element has a multiplicative inverse, i.e., every nonzero element is a unit. Example

1

Q, R, and C are fields.

2

Zp are fields if and only if p is a prime number.

3

Z, F[x] are not fields.

Instructor: Yifan Yang Review on the basic properties of fields

Fields

Definition A field F is a commutative ring with unity 1 = 0 such that every nonzero element has a multiplicative inverse, i.e., every nonzero element is a unit. Example

1

Q, R, and C are fields.

2

Zp are fields if and only if p is a prime number.

3

Z, F[x] are not fields.

Instructor: Yifan Yang Review on the basic properties of fields

Fields

Definition A field F is a commutative ring with unity 1 = 0 such that every nonzero element has a multiplicative inverse, i.e., every nonzero element is a unit. Example

1

Q, R, and C are fields.

2

Zp are fields if and only if p is a prime number.

3

Z, F[x] are not fields.

Instructor: Yifan Yang Review on the basic properties of fields

Characteristic of a ring

Definition Let R be a ring. Suppose that there is a positive integer n such that n · a = a + · · · + a = 0 for all a ∈ R. Then the smallest such integer n is the characteristic of R. If no such an integer exists, then the characteristic is 0. Example

1

Z, Q, R, and C are all of characteristic 0.

2

Zn has characteristic n.

Instructor: Yifan Yang Review on the basic properties of fields

Characteristic of a ring

Definition Let R be a ring. Suppose that there is a positive integer n such that n · a = a + · · · + a = 0 for all a ∈ R. Then the smallest such integer n is the characteristic of R. If no such an integer exists, then the characteristic is 0. Example

1

Z, Q, R, and C are all of characteristic 0.

2

Zn has characteristic n.

Instructor: Yifan Yang Review on the basic properties of fields

Characteristic of a ring

Definition Let R be a ring. Suppose that there is a positive integer n such that n · a = a + · · · + a = 0 for all a ∈ R. Then the smallest such integer n is the characteristic of R. If no such an integer exists, then the characteristic is 0. Example

1

Z, Q, R, and C are all of characteristic 0.

2

Zn has characteristic n.

Instructor: Yifan Yang Review on the basic properties of fields

Characteristic of a field

Theorem (Exercise 19.29) Let D be an integral domain. Then the characteristic of D is either 0 or a prime number p. Corollary If n is a composite number, then Zn is not a field. Also, if F is a finite field, then the number of elements in F is pn for some n, where p is the characteristic of F. Warning In future exams, you will be asked to construct fields of pk

• elements. If you give Zpk as answers, you will receive double

penalty!! I.e., not only will you lose the points on the problem, the same number of points will also be deducted from your scores obtained from other problems.

Instructor: Yifan Yang Review on the basic properties of fields

Characteristic of a field

Theorem (Exercise 19.29) Let D be an integral domain. Then the characteristic of D is either 0 or a prime number p. Corollary If n is a composite number, then Zn is not a field. Also, if F is a finite field, then the number of elements in F is pn for some n, where p is the characteristic of F. Warning In future exams, you will be asked to construct fields of pk

• elements. If you give Zpk as answers, you will receive double

penalty!! I.e., not only will you lose the points on the problem, the same number of points will also be deducted from your scores obtained from other problems.

Instructor: Yifan Yang Review on the basic properties of fields

Characteristic of a field

Theorem (Exercise 19.29) Let D be an integral domain. Then the characteristic of D is either 0 or a prime number p. Corollary If n is a composite number, then Zn is not a field. Also, if F is a finite field, then the number of elements in F is pn for some n, where p is the characteristic of F. Warning In future exams, you will be asked to construct fields of pk

• elements. If you give Zpk as answers, you will receive double

penalty!! I.e., not only will you lose the points on the problem, the same number of points will also be deducted from your scores obtained from other problems.

Instructor: Yifan Yang Review on the basic properties of fields

Prime fields

Theorem (27.19) If a field is of finite characteristic p, then it contains a subfield isomorphic to Zp. If a field is of characteristic 0, then it contains a subfield isomorphic to Q. Definition The fields Zp and Q are called prime fields.

Instructor: Yifan Yang Review on the basic properties of fields

Prime fields

Theorem (27.19) If a field is of finite characteristic p, then it contains a subfield isomorphic to Zp. If a field is of characteristic 0, then it contains a subfield isomorphic to Q. Definition The fields Zp and Q are called prime fields.

Instructor: Yifan Yang Review on the basic properties of fields

Multiplicative group of F ×

Corollary (23.6) If G is a finite subgroup of the multiplicative group F × of a field F, then G is cyclic. In particular, if F is a finite field, then F × is a cyclic group.

Instructor: Yifan Yang Review on the basic properties of fields

Maximal ideals and fields

Theorem (27.9) Let R be a commutative ring with unity, and I be an ideal. Then R/I is a field if and only if I is a maximal ideal. Theorem (45.12) An ideal p in a PID is a maximal ideal if and only if p is an irreducible. Example

1

Z[i]/2 + i and Z[i]/3 are fields of 5 and 9 elements, respectively.

2

Z[ √ −5]/1 + √ −5 is not a field, even though 1 + √ −5 is an irreducible. (Note that Z[ √ −5] is not a PID.)

Instructor: Yifan Yang Review on the basic properties of fields

Maximal ideals and fields

Theorem (27.9) Let R be a commutative ring with unity, and I be an ideal. Then R/I is a field if and only if I is a maximal ideal. Theorem (45.12) An ideal p in a PID is a maximal ideal if and only if p is an irreducible. Example

1

Z[i]/2 + i and Z[i]/3 are fields of 5 and 9 elements, respectively.

2

Z[ √ −5]/1 + √ −5 is not a field, even though 1 + √ −5 is an irreducible. (Note that Z[ √ −5] is not a PID.)

Instructor: Yifan Yang Review on the basic properties of fields

Maximal ideals and fields

Theorem (27.9) Let R be a commutative ring with unity, and I be an ideal. Then R/I is a field if and only if I is a maximal ideal. Theorem (45.12) An ideal p in a PID is a maximal ideal if and only if p is an irreducible. Example

1

Z[i]/2 + i and Z[i]/3 are fields of 5 and 9 elements, respectively.

2

Z[ √ −5]/1 + √ −5 is not a field, even though 1 + √ −5 is an irreducible. (Note that Z[ √ −5] is not a PID.)

Instructor: Yifan Yang Review on the basic properties of fields

Maximal ideals and fields

Theorem (27.9) Let R be a commutative ring with unity, and I be an ideal. Then R/I is a field if and only if I is a maximal ideal. Theorem (45.12) An ideal p in a PID is a maximal ideal if and only if p is an irreducible. Example

1

Z[i]/2 + i and Z[i]/3 are fields of 5 and 9 elements, respectively.

2

Z[ √ −5]/1 + √ −5 is not a field, even though 1 + √ −5 is an irreducible. (Note that Z[ √ −5] is not a PID.)

Instructor: Yifan Yang Review on the basic properties of fields

Maximal ideals and fields

Theorem (27.24) If F is a field, then F[x] is a PID. Remark The above theorems provide an easy method to construct new

• fields. Namely, pick an irreducible polynomial p(x) over F, and

then we have a field F[x]/p(x).

Instructor: Yifan Yang Review on the basic properties of fields

Maximal ideals and fields

Theorem (27.24) If F is a field, then F[x] is a PID. Remark The above theorems provide an easy method to construct new

• fields. Namely, pick an irreducible polynomial p(x) over F, and

then we have a field F[x]/p(x).

Instructor: Yifan Yang Review on the basic properties of fields