Self-reciprocal Polynomials Over Finite Fields

Helmut Meyn ; Werner Götz

Séminaire lotharingien de combinatoire, Tome 21 (1989)

Résumé

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 F_q. 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 f^Q of degree 2n given by f^Q (x) := xⁿ f(x + x^-1). The criterion states that the self-reciprocal polynomial f^Q 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].