A study of monogenity of binomial composition
Volume 221 / 2025
Abstract
Let $\theta $ be a root of a monic polynomial $h(x) \in {\mathbb Z}[x]$ of degree $n \geq 2$. We say that $h(x)$ is monogenic if it is irreducible over ${\mathbb Q}$ and $\{ 1, \theta , \theta ^2, \ldots , \theta ^{n-1} \}$ is a basis for the ring $\mathbb{Z} _K$ of integers of $K = {\mathbb Q}(\theta )$. We investigate monogenity of number fields generated by roots of compositions of two binomials. We characterise all the primes dividing the index of the subgroup ${\mathbb Z}[\theta ]$ in ${\mathbb Z}_K$ where $K = {\mathbb Q}(\theta )$ with $\theta $ having minimal polynomial $F(x) = (x^m-b)^n - a \in {\mathbb Z}[x]$, $m\geq 1$ and $n \geq 2$. 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.
