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8.Suppose p is a prime and that f and g are monic polynomials over Z, (that is, the leading coefficients are 1) of degree m, and that the congruence f(x)

= g(x) mod p has at least m distinct solutions mod p. Show that the polynomials are identical. Use this to prove Wilson's Theorem, that if p is an odd prime then (p-1)! = -1 mod p.

Fig: 1