On Sets of Irreducible Polynomials Closed by Composition

• Published in: Arithmetic of Finite Fields Vol. 10064; pp. 77 - 83
• Main Authors: Ferraguti, Andrea, Micheli, Giacomo, Schnyder, Reto
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2017; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Finite fields; Graphs; Irreducible polynomials; Semigroups
• ISBN: 3319552260; 9783319552262
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-55227-9_6

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}$$\end{document} be a set of monic degree 2 polynomials over a finite field and let C be the compositional semigroup generated by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal S$$\end{document}. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}$$\end{document}. 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.