information:filtering, linguistic:paraphrasing
Polynomials over a finite field

Let F be a finite field with q = p elements, where p is an odd prime number and n is positive, and let F be the ring of polynomials in one variable with coefficients in F. If f, g ∈ F {\displaystyle f,g\in F} and f is irreducible, monic, and has positive degree, define the quadratic character for F in the usual manner:

If f = f 1 ⋯ f n {\displaystyle f=f_{1}\cdots f_{n}} is a product of monic irreducibles let

Dedekind proved that if f, g ∈ F {\displaystyle f,g\in F} are monic and have positive degrees,