On the action of \(\text{GL}_2(\mathbb F_q)\) on irreducible polynomials over \(\mathbb F_q\)
The author shows that \(\text{GL}_2(\mathbb F_q)\) acts on the set of irreducible polynomials over \(\mathbb F_q\) of degree at least 2. This action is studied and the cardinality of the set of fixed points of certain subgroups of \(\text{GL}_2(\mathbb F_q)\) is computed. The results include enumeration of irreducible polynomials that are invariant under the substitution \(X\rightarrow X + b\) and under the substitution \(X\rightarrow aX\). One of the main objectives is the characterization of the sets of fixed points of the action of single elements of \(\text{GL}_2(\mathbb F_q)\) of order \(r\), where \(r\) is a prime divisor of \(q\) or \(q-1\) (or equivalently of cyclic subgroups) on the set \(I_n\) of monic irreducibles of degree \(n\), and their cardinalities are computed. In fact it is shown that the set of fixed points of the action of any element of a \(p\)-Sylow subgroup of \(\text{GL}_2(\mathbb F_q)\), where \(p\) is the characteristic of \(F_q\), relates to irreducibles that are invariant under the substitution \(X\rightarrow X + b\). The number of even irreducible polynomials of given degree is obtained. The results are combined to provide enumeration formulas for the fixed point of upper triangular matrices.
scientific article; zbMATH DE number 5886343