MA3412S2_Hil2014.pdf

# 1 2 19 an integer z is said to be a quadratic residue

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An integer z is said to be a quadratic residue of a prime number p if there exists some integer w such that z w 2 (mod p ). Now the non-zero elements of any field constitute a group under multiplication, and every finite subgroup of the group of non-zero elements of a field is cyclic. This result can be applied to the field of congruence classes of integers modulo p to deduce that, given any prime number p , there exists some in- teger g whose congruence class generates the group of non-zero elements of this field. Then, given any integer z coprime to p , there exists some integer k such that z g k (mod p ). Such an integer g is said to be a primitive root of p . In the case where p 6 = 2 the group of congruence classes modulo p of integers coprime to p is of even order and it follows from this that an integer z is a quadratic residue of p if and only if z g k (mod p ) for some even integer k , where g is some primitive root of p , and this is the case if and only if z ( p - 1) / 2 1 (mod p ). In particular, - 1 is a quadratic residue of p if and only if ( - 1) ( p - 1) / 2 1 (mod p ). It follows that - 1 is a quadratic residue of an odd prime p if and only if p 1 (mod 4). 2.5 Maximal Ideals and Prime Ideals Definition Let R be a unital commutative ring. An ideal M of R is said to be maximal if it is a proper ideal of R and if the only ideals I satisfying M I R are the maximal ideal M and the ring R itself. Lemma 2.13 An ideal M of a unital commutative ring R is a maximal ideal of R if and only if the quotient ring R/M is a field. Proof The preimage ν - 1 ( J ) of any ideal of R/M under the quotient homo- morphism ν : R R/M is an ideal of R satisfying M ν - 1 ( J ) R . Also each ideal I of R satisfying M I R determines an ideal I/M of R/M satisfying ν - 1 ( I/M ) = I , and moreover an ideal J of R/M satisfies J = I/M if and only if ν - 1( J ) = I . Thus the ideals I of R that satisfy M I R are in one-to-one correspondence with the ideals of the quotient ring R/M . It follows that an ideal M of R is maximal if and only if the only ideals of the quotient ring R/M are the zero ideal and the whole of the quotient ring. The quotient R/M of a unital commutative ring R by a proper ideal M is a commutative ring with a non-zero multiplicative identity element M + 1 R . But a unital commutative ring is a field if and only if the only ideals of that ring are the zero ideal and the ring itself. (Lemma 1.4). It follows that an ideal M of a unital commutative ring R is a maximal ideal of R if and only if the corresponding quotient ring R/M is a field. 20
Definition Let R be a unital commutative ring. An ideal P of R is said to be prime if P is a proper ideal of R and, for all elements x and y of R satisfying xy P , either x P or y P .

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• Fall '16
• Jhon Smith
• Algebra, Integers, Prime number, Integral domain, Ring theory, Principal ideal domain

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