A Study of monogenity of Binomial Composition

Abstract

Let $\theta$ be a root of a monic polynomial $h(x)\in \mathbb{Z}[x]$ of degree $n\ge 2$. We say $h(x)$ is monogenic if it is irreducible over $\Q$ and $\{1,\theta ,{\theta }^2,\dots ,{\theta }^{n-1}\}$ is a basis for the ring ${\mathbb{Z}}_K$ of integers of $K = \Q(θ)$. In this article, we study about the monogenity of number fields generated by a root of composition of two binomials. We characterise all the primes dividing the index of the subgroup $\mathbb{Z}[\theta ]$ in ${\mathbb{Z}}_K$ where $K = \Q(θ)$ with $\theta$ having minimal polynomial $F(x)=(x^m-b{)}^n-a\in \mathbb{Z}[x]$, $m\ge 1$ and $n\ge 2$. As an application, we provide a class of pairs of binomials $f(x)=x^n-a$ and $g(x)=x^m-b$ having the property that both $f(x)$ and $f(g(x))$ are monogenic.

9 pages