An outline for the proof of Theorem 17.11 follows.
(a) Prove that the operations defined in part (a) of Theorem 17.11 are well-defined by showing that if f(x) = f1(x) (mods(x)) and g(x) = g1(x) (mod s(x)), then f(x) + g(x) = f1(x) + g1(x) (mods(x))
(b) Verify the ring properties for the equivalence classes in F[x](s(x)).
(c) Let f(x) ∈ F[x], with f(x) ≠ 0 and degree f(x) < degree 5 (x). If s(x) is irreducible in F[x], why does it follow that 1 is the gcd of f(x) and s(x)?
(d) Use part (c) to prove that if s(x) is irreducible in F[x], then F[x]/(s(x) is a field.
(e) If |F| = q and degree s(x) = n, determine the order of F[x]/(s(x)).