On commutative semigroups of polynomials with respect to composition (Q1076079)

This paper considers commutative semigroups of Dickson-polynomials with respect to composition over integral domains. It is proved that the set of all Dickson-polynomials \(\{g_ k(r,x)| k\in {\mathbb{N}}\}\) over an integral domain D is closed under composition iff \(r=0\) or \(r=1\). Also the set of all Dickson-polynomials of even degree \(\{g_ k(r,x)| k\in 2{\mathbb{N}}\}\) over D is closed under composition iff \(r=0\) or \(r=1\). The set of all Dickson-polynomials of odd degree \(\{g_ k(r,x)| k\in 2{\mathbb{N}}-1\}\) over D is closed under composition iff \(r=0\), \(r=1\) or \(r=- 1\). - Subsequently all commutative semigroups of polynomials which contain only polynomials of positive even degree and at least one polynomial of each positive even degree over a field of characteristic different from 2 and 3 are determined. Further, all commutative semigroups which contain only polynomials of odd degree and at least one polynomial of each odd degree over fields of characteristic different from 3 and 5 are characterized. Where appropriate the corresponding results for polynomials over integral domains are given.