A Note on Abelian Monogenic Trinomials

Abstract

An abelian monogenic polynomial $f(x)\in \mathbb{Z}[x]$ is a monic polynomial of degree $N$ that is irreducible over $\mathbb{Q}$, such that the Galois group of $f(x)$ over $\mathbb{Q}$ is abelian, and $\{1,\hat{I}¸,\hat{I}{¸}^2,\dots ,\hat{I}{¸}^{N-1}\}$ is a basis for the ring of integers of $\mathbb{Q}(\hat{I}¸)$, where $f(\hat{I}¸)=0$. In this article, we determine all abelian monogenic trinomials of the form $x^{2n}+a{x}^n+b$, where $n,a,b\in \mathbb{Z}$ with $n\ge 1$ and $ab\ne 0$.