On the reducibility behaviour of Thue polynomials

TL;DR

This paper proves that, with few exceptions, Thue polynomials over the rationals have only finitely many reducible integer specializations, using ramification theory and permutation group classification.

Contribution

It establishes a general result confirming a conjecture about the reducibility of Thue polynomials, extending previous special case proofs.

Findings

Most Thue polynomials have finitely many reducible specializations

The proof relies on permutation group classification and ramification theory

Few explicit exceptions are identified

Abstract

We prove a result about reducibility behaviour of Thue polynomials over the rationals that was conjectured by M\"uller. More precisely, we show that, apart from few explicitly given exceptions, these polynomials have only finitely many reducible integer specializations. Special cases have been proved e.g. by M\"uller and Langmann. The proof uses ramification theory to reduce the assertion to a statement about permutation groups containing an $n$-cycle. This statement is finally proven with the help of the classification of primitive permutation groups containing an $n$-cycle (a result which rests on the classification of finite simple groups).