A Study of monogenity of Binomial Composition

Abstract

Let $θ$ be a root of a monic polynomial $h(x) \in \Z[x]$ of degree $n \geq 2$. We say $h(x)$ is monogenic if it is irreducible over $\Q$ and $\{ 1, θ, θ^2, \ldots, θ^{n-1} \}$ is a basis for the ring $\Z_K$ 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 $\Z[θ]$ in $\Z_K$ where $K = \Q(θ)$ with $θ$ having minimal polynomial $F(x) = (x^m-b)^n - a \in \Z[x]$, $m\geq 1$ 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.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 1.4
• Corollary 1.5
• Corollary 1.6
• Example 1.7
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3