Factorization of polynomials over finite fields
Richard G. Swan · Pacific Journal of Mathematics · 1962
Dickson [1, Ch.V, Th. 38] has given an interesting necessary condition for a polynomial over a finite field of odd characteristic to be irreducible.In Theorem 1 below, I will give a generalization of this result which can also be applied to fields of characteristic 2. It also applies to reducible polynomials and gives the number of irreducible factors mod 2.Applying the theorem to the polynomial x p -1 gives a simple proof of the quadratic reciprocity theorem.Since there is some interest in trinomial equations over finite fields, e.g.[2], [4], I will also apply the theorem to trinomials and so determine the parity of the number of irreducible factors.where a lf , a n are the roots of f(x) (counted with multiplicity) in some extension field of F. Clearly D(/) = 0 if / has any repeated τoot.Since D(f) is a symmetric function in the roots of /, D(f) e F.An alternative formula for D(f) which is sometimes useful may be obtained as follows:where n is the degree of f(x) and f\x) the derivative of f(x).In § 4, I will give still another way to calculate D(f).If f(x) is monic with integral coefficients in some p-adic or algebraic number field, all a t are integral and so D(f) is integral.Consider the expression This is integral and lies in F, being a symmetric function of the roots.Clearly δ(/) = 8 X + 2δ 2 where δ 2 is integral.Thus D(f) = δ(/) 2 = dl mod 4, so D(f) is congruent to a square in F mod 4.This is a special case of a well-known theorem of Stickelberger [3, Ch. 10, Sec.3].