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 Q , R , and C are fields. 1 Z p are fields if and only if p is a prime number. 2 Z , F [ x ] are not fields. 3 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 Q , R , and C are fields. 1 Z p are fields if and only if p is a prime number. 2 Z , F [ x ] are not fields. 3 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 Q , R , and C are fields. 1 Z p are fields if and only if p is a prime number. 2 Z , F [ x ] are not fields. 3 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 Q , R , and C are fields. 1 Z p are fields if and only if p is a prime number. 2 Z , F [ x ] are not fields. 3 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 Z , Q , R , and C are all of characteristic 0. 1 Z n has characteristic n . 2 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 Z , Q , R , and C are all of characteristic 0. 1 Z n has characteristic n . 2 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 Z , Q , R , and C are all of characteristic 0. 1 Z n has characteristic n . 2 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 Z n is not a field. Also, if F is a finite field, then the number of elements in F is p n for some n , where p is the characteristic of F . Warning In future exams, you will be asked to construct fields of p k elements. If you give Z p k 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 Z n is not a field. Also, if F is a finite field, then the number of elements in F is p n for some n , where p is the characteristic of F . Warning In future exams, you will be asked to construct fields of p k elements. If you give Z p k 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 Z n is not a field. Also, if F is a finite field, then the number of elements in F is p n for some n , where p is the characteristic of F . Warning In future exams, you will be asked to construct fields of p k elements. If you give Z p k 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 Z p . If a field is of characteristic 0 , then it contains a subfield isomorphic to Q . Definition The fields Z p 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 Z p . If a field is of characteristic 0 , then it contains a subfield isomorphic to Q . Definition The fields Z p 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 Z [ i ] / � 2 + i � and Z [ i ] / � 3 � are fields of 5 and 9 elements, 1 respectively. √ √ √ Z [ − 5 ] / � 1 + − 5 � is not a field, even though 1 + − 5 is 2 √ 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 Z [ i ] / � 2 + i � and Z [ i ] / � 3 � are fields of 5 and 9 elements, 1 respectively. √ √ √ Z [ − 5 ] / � 1 + − 5 � is not a field, even though 1 + − 5 is 2 √ 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 Z [ i ] / � 2 + i � and Z [ i ] / � 3 � are fields of 5 and 9 elements, 1 respectively. √ √ √ Z [ − 5 ] / � 1 + − 5 � is not a field, even though 1 + − 5 is 2 √ 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 Z [ i ] / � 2 + i � and Z [ i ] / � 3 � are fields of 5 and 9 elements, 1 respectively. √ √ √ Z [ − 5 ] / � 1 + − 5 � is not a field, even though 1 + − 5 is 2 √ 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
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