Let S be a set of monic degree 2 polynomials over a finite field and let C be the compositional semigroup generated by S. In this paper we establish a necessary and sufficient condition for C to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set S. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree 2 polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions.
|Titolo:||On Sets of Irreducible Polynomials Closed by Composition|
|Titolo del libro:||Arithmetic of Finite Fields. WAIFI 2016.|
|Data di pubblicazione:||2017|
|Settore Scientifico Disciplinare:||Settore MAT/03 - Geometria|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1007/978-3-319-55227-9_6|
|Appare nelle tipologie:||2.1 Contributo in volume (Capitolo o Saggio)|