What is the Proof for (p-1)!≡±1 (mod p) in Number Theory?

[koukou]

Recall the definition of n! (read n factorial"):
n! = (n)(n-1)(n-2) ….(2)(1) =∏(k)
In both (a) and (b) below, suppose p≥3 is prime.
(a) Prove that if x∈ Zpx is a solution to x square ≡1 (mod p), then x ≡±1 (mod p).
(b) Prove that (p-1)!≡±1 (mod p)

Zpx x shoud be above p

a and b looks like some theorem proof

 

[MathematicalPhysicist]

Hints:
1. factorise x^2-1=(x-1)(x+1).

2. this theorem is called wilson theorem.

 

[robert Ihnot]

Hint: (2) can be looked at as a case of a and its inverse. The first part, (1) plays a special role in that.