Using Exercise 27, deduce the half of Wilson's theorem that states that if p is a prime, then (p - 1)! ≡ -1 (mod p ). [The other half states that if n is an integer > 1 such that (n - 1 )! ≡ -1 (mod n ), then n is a prime. Just think what the remainder of (n - l)! would be modulo n if n is not a prime.]

Data from Exercise 27

Show that 1 and p - 1 are the only elements of the field Z_P that are their own multiplicative inverse.