An integral domain Z is a ring for the operations + and * with three additional properties:

1.  The commutative property of *: For any elements x and y in Z, x*y=y*x.

2.  The unity property: There is an element 1 in Z that is the identity for *, meaning for any z in Z, z*1=z. Also, 1 has to be shown to be different from the identity of +.

3.  The no zero divisors property: For any two elements a and b in Z both different from the identity of +, a*b?0.

A field F is an integral domain with the additional property that for every element x in F that is not the identity under +, there is an element y in F so that x*y=1 (1 is notation for the unity of an integral domain). The element y is called the multiplicative inverse of x. Another way to explain this property is that multiplicative inverses exist for every nonzero element.

Modular multiplication, [*], is defined in terms of integer multiplication by this rule: [a]m [*] [b]m = [a * b]m

Note: For ease of writing notation, follow the convention of using just plain * to represent both [*] and *. Be aware that one symbol can be used to represent two different operations (modular multiplication versus integer multiplication).

A.  Prove that the ring Z31 (integers mod 31) is an integral domain by using the definitions given above to prove the following are true:

1.  The commutative property of [*]

2.  The unity property

3.  The no zero divisors property

B.  Prove that the integral domain Z31 (integers mod 31) is a field by using the definition given above to prove the existence of a multiplicative inverse for every nonzero element