Galois Groups of Trinomials

Abstract

Galois groups of irreducible trinomials X n + aX s + b ∈ X are investigated assuming the classification of finite simple groups. We show that under some simple yet general hypotheses bearing on the integers n s a and b only very specific groups can occur. For instance, if the two integers nb and as n − s are coprime and if s is a prime number, then already the Galois group of f X is either the alternating group A n or the symmetric group S n . This significantly extends work of Osada.

A Class of Trinomials with Galois Group Sn

“…+ · · · + x + 1 contains A n , the alternating group on n letters for every n (see [12]). In this paper, we extend the above mentioned result of Schur to polynomials f n (x) defined by (1). Precisely stated, we prove: Theorem 1.1.…”

“…By Hensel's Lemma [8,Chap. II,4.6], f (x) factors over the field Q q of q-adic numbers into monic polynomials as f 1 …”

“…+ · · · + a 1 x + a 0 , a i ∈ Z, (a 0 a n , n!) = 1 (1) is irreducible over the field Q of rational numbers. In 1930, he further showed that the Galois group of the nth Taylor polynomial of the exponential function given by…”

“…Examples of trinomials with Galois group S n occur in [1], [7] and [9]. These do not cover the class of trinomials given by the following theorems; moreover our method of constructing such trinomials is quite different.…”

Dynamical Galois groups of trinomials and Odoni's conjecture

“…It is also worth remarking that the topic of Galois groups of trinomials over has been extensively studied (see, for example, ). Hilbert used trinomials to prove the existence, for each degree , of a polynomial over whose Galois group is .…”