If px is a nonzero constant polynomial in Fx show that any t

If p(x) is a nonzero constant polynomial in F[x], show that any two polynomials in F[x] are congruent modulo p(x).

Solution

Let f(x) and g(x) be two polynomials in F[x] and note because p(x) is constant, p(x) = p for some p F. Then p 1 is in F, and hence p 1 (f(x) g(x)) F[x]. Thus p(x) | (f(x) g(x)), because p · p 1 (f(x) g(x)) = (f(x) g(x)). We conclude that f(x) g(x) (mod p) as required.