Periods of maps on irreducible polynomials over finite fields (Q1266430)

A polynomial \(\sigma(x)\) over a field \(F\) defines a dynamical system on the algebraic closure of \(F\). It is expedient to study the action of \(\sigma(x)\) on irreducible polynomials rather than on field elements: \(\sigma(x)\) maps an irreducible polynomial \(g(x)\) over \(F\) to the minimal polynomial of \(\sigma(\alpha)\), where \(\alpha\) is a root of \(g(x)\). In the case where \(F\) is the finite field \(F_q\), and \(\sigma(x)=x^q+ax\) (\(a\) is a non-zero element of \(F_q\)), the author proves that for every positive integer \(n\) there are infinitely many polynomial periodic orbits of minimal period \(n\). The proof highlights the dynamical significance of the so-called Carlitz module.