Solution for Section G.3 Question 2

2. We must check that the following properties hold, where + is addition modulo p and · is multiplication modulo p.

For (Z_p,+):

• closure
  Since the sum of any two elements of Z_p is reduced modulo p, we obtain an element of Z_p. So Z_p is closed under addition modulo p.
  
• associativity
  Addition is associative so addition modulo p is associative.
  
• commutativity
  Addition is commutative so addition modulo p is commutative.
  
• identity
  The element [0] is the (additive) identity, since [0] + [a] = [0] for all [a] in Z_p.
  
• inverse
  The element [-a] is the (additive) inverse of the element [a], since [-a] + [a] = [0].
  

For (Z_p-{0}, ·):

• closure, associativity, identity and inverse
  These were shown in Section G.1 Question 6.
  
• commutativity
  Multiplication is commutative so multiplication modulo p is commutative.
  

For all a,b,c in Z_p, a · (b + c) = a · b + a · c
The statement is true. Multiplication modulo p distributes over addition modulo p.

Back to Section G.3