Necklaces, symmetries and self-reciprocal polynomials

Abstract

The connection between a certain class of necklaces and self-reciprocal polynomials over finite fields is shown. For n⩾2, self-reciprocal polynomials of degree 2n arising from monic irreducible polynomials of degree n are shown to be either irreducible or the product or two irreducible factors which are necessarily reciprocal polynomials. Using DeBruijn's method we count the number of necklaces in this class and hence obtain a formula for the number of irreducible self-reciprocal polynomials showing that they exist for every even degree. Thus every extension of a finite field of even degree can be obtained by adjoining a root of an irreducible self-reciprocal polynomial.