[Formerly: Publ. I.R.M.A. Strasbourg, 1990, 413/S-21, p. 82-90.]

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 *F _{2}*,
as given
by Varshamov and Garakov in [Var69]. Any polynomial

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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