On different families of invariant irreducible polynomials over [formula omitted

Abstract

Using a natural action of the permutation group S 3 on the set of irreducible polynomials, we attach to each subgroup of S 3 the family of its invariant polynomials. Enumeration formulas for the trivial subgroup and for one transposition subgroup were given by Gauss (1863) (for prime fields) [1] and Carlitz (1967) (for all finite base fields) [2]. Respectively, they allow to enumerate all irreducible and self-reciprocal irreducible polynomials. In our context, the last remaining case concerned the alternating subgroup A 3 . We give here the corresponding enumeration formula restricted to F 2 base field. We wish this will give an interesting basis for subsequent developments analogous to those of Meyn (1990) [3] and Cohen (1992) [4].