Exceptional polynomials with 2-transitive affine monodromy groups
type: Item
instance of: scholarly article
title: Exceptional polynomials with 2-transitive affine monodromy groups
zbMATH Open document ID: 1264.12003
author: Florian Möller
published in: Finite Fields and their Applications
publication date: 2012-03-22T00:00:00Z
review text: Let \(k\) denote a finite field with algebraic closure \(K\). A polynomial \(f(X)\in k[X]\) is called exceptional if \(K\) contains infinitely many finite extensions of \(k\), each of whose elements are permuted by \(f\). Let \(t\) denote a transcendental over \(k\). The main result of the paper under review is the classification of exceptional polynomials \(f(X)\) such that the polynomial \(f(X)-t\in k(t)[X]\) is separable and its Galois group \(A\) (the affine arithmetic monodromy group) acts doubly transitively on its roots. It is shown that these polynomials \(f\) belong to well-known families of exceptional polynomials. In particular, in the Galois group \(G\) of \(f(X)-t\) over \(K(t)\) (the affine geometric monodromy group) the stabilizer of any root is cyclic or dihedral. Such polynomials \(f\) are constructed explicitly in the paper.
reviewed by: Péter E. Frenkel
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 12E05
zbMATH DE Number: 6017555
zbMATH Keywords: exceptional polynomials
zbMATH Keywords: permutation polynomials
zbMATH Keywords: monodromy groups
zbMATH Keywords: Riemann--Hurwitz genus formula
MaRDI profile type: Publication
full work available at URL: j.ffa.2011.10.003
OpenAlex ID: W2050754091
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DOI: 10.1016/J.FFA.2011.10.003
description: scientific article; zbMATH DE number 6017555