Count the elements of F[x]/(p(x)) where F is a finite field

Let F be a finite field of order q and let p(x) be a polynomial in F[x] of degree n \geq 1. Show that F[x]/(p(x)) has order q^n.

In this previous exercise, we showed that every coset in F[x]/(p(x)) is represented by a polynomial of degree less than n, and that two such polynomials represent distinct cosets. Evidently then the number of elements of F[x]/(p(x)) is precisely the number of polynomials in F[x] having degree less than n. Each such polynomial is uniquely determined by its coefficients, of which there are n, each taking one of the q values in F. Thus F[x]/(p(x)) contains q^n elements.

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