The monogenicity and Galois groups of certain reciprocal quintinomials

Abstract

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for ${\mathbb Z}_K$, the ring of integers of $K={\mathbb Q}(θ)$, where $f(θ)=0$. For $n\ge 2$, we define the reciprocal quintinomial \[{\mathcal F}_{n,A,B}(x):=x^{2^n}+Ax^{3\cdot 2^{n-2}}+Bx^{2^{n-1}}+Ax^{2^{n-2}}+1\in {\mathbb Z}[x].\] In this article, we extend our previous work on the monogenicity of ${\mathcal F}_{n,A,B}(x)$ to treat the specific previously-unaddressed situation of $A\equiv B\equiv 1\pmod{4}$. Moreover, we determine the Galois group over ${\mathbb Q}$ of ${\mathcal F}_{n,A,B}(x)$ in special cases.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 2.4: Dedekind Cohen
• Theorem 2.5
• Remark 2.6
• Definition 2.7
• Corollary 2.8