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On the monogenicity and Galois groups of $\boldsymbol{x^{2p}+ax^p+b^p}$

Joshua Harrington, Lenny Jones

Abstract

Let $f(x)=x^{2p}+ax^p+b^p$, where $p$ is a prime and $a,b\in {\mathbb Z}$ with $ab\ne 0$. If $f(x)$ is irreducible over ${\mathbb Q}$, we say that $f(x)$ is monogenic if $\{1,θ,θ^2,\ldots ,θ^{2p-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. In this article, we give a characterization of the monogenic trinomials $f(x)$ according to their Galois groups. These results extend prior investigations of the authors.

On the monogenicity and Galois groups of $\boldsymbol{x^{2p}+ax^p+b^p}$

Abstract

Let , where is a prime and with . If is irreducible over , we say that is monogenic if is a basis for the ring of integers of , where . In this article, we give a characterization of the monogenic trinomials according to their Galois groups. These results extend prior investigations of the authors.
Paper Structure (4 sections, 6 theorems, 14 equations)

This paper contains 4 sections, 6 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Proof of Theorem \ref{['Thm:Main']}
  4. The Proof of Corollary \ref{['Cor:Main']}

Key Result

Theorem 1.1

Suppose that $f(x)$, as defined in Eq:f, is irreducible over ${\mathbb Q}$, and let $C_N$ denote the cyclic group of order $N$. Then

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['Thm:Main']}
  • proof