Self-reciprocal Polynomials Over Finite Fields

The reciprocal f ∗(x) of a polynomial f(x) of degree n is defined by f ∗(x) = xf(1/x). A polynomial is called self-reciprocal if it coincides with its reciprocal. The aim of this paper is threefold: first we want to call attention to the fact that the product of all self-reciprocal irreducible monic (srim) polynomials of a fixed degree has structural properties which are very similar to those of the product of all irreducible monic polynomials of a fixed degree over a finite field Fq. In particular, we find the number of all srim-polynomials of fixed degree by a simple Möbius-inversion. The second and central point is a short proof of a criterion for the irreducibility of self-reciprocal polynomials over F2, as given by Varshamov and Garakov in [10]. Any polynomial f of degree n may be transformed into the self-reciprocal polynomial f of degree 2n given by f(x) := xnf(x+x−1). The criterion states that the self-reciprocal polynomial f is irreducible if and only if the irreducible polynomial f satisfies f ′(0) = 1. Finally we present some results on the distribution of the traces of elements in a finite field. These results were obtained during an earlier attempt to prove the criterion cited above and are of some independent interest. For further results on self-reciprocal polynomials see the notes of chapter 3, p. 132 in Lidl/Niederreiter [5]. 1 The rôle of the polynomial x n+1 − 1 Some remarks on self-reciprocal polynomials are in order before we can state the main theorem of this section. • If f is self-reciprocal then the set of roots of f is closed under the inversion map α 7→ α−1 (α 6= 0). Universität Erlangen-Nürnberg, Informatik I, Martensstr. 3, D-91058 Erlangen