Séminaire Lotharingien de Combinatoire, B21d (1989), 8 pp.
[Formerly: Publ. I.R.M.A. Strasbourg, 1990, 413/S-21, p. 82-90.]

Helmut Meyn and Werner Götz

Self-reciprocal Polynomials Over Finite Fields

Abstract. The reciprocal f*(x) of a polynomial f(x) of degree n is defined by f*(x) =xnf(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 [Var69]. Any polynomial f of degree n may be transformed into the self-reciprocal polynomial fQ of degree 2n given by fQ (x) := xn f(x + x-1). The criterion states that the self-reciprocal polynomial fQ 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 [Lid83].

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