10. Fermat's theorem

(Z*n, *n, [1]n).

(a) Show that

([a]n) = {[1]n} ∪ {[a]m | m ∈ N}

for any [a]n ∈ Z*n.

(b) For a ∈ Z with gcd

(a, n) = 1, argue that there must exist a smallest number / in

[0] ∪ N such that [a]'n = [1]n.

(c) Use Lagrange's theorem to conclude that there exists some k ∈ N such that k ⋅ l = φ(n)

for the minimal l of part (b).

(d) Use Corollary 2.27 and the equation k ⋅ l = φ(p) to show that [a^(p-1)]p = [1]p.

Explain why this proves Fermat's theorem.