ON NONMONOGENIC ALGEBRAIC NUMBER FIELDS

Abstract

Let $q$ be a prime number and $f(x)=x^{q^s}-a{x}^m-b$ be a monic irreducible polynomial of degree $q^s$ having integer coefficients. Let $K=\mathbb{Q}(𝜃)$ be an algebraic number field with $𝜃$ a root of $f(x)$. We give some explicit conditions involving only $a,b,m,q,s$ for which $K$ is not monogenic. As an application, we show that if $p$ is a prime number of the form $32k+1$, $k\in \mathbb{Z}$ and $𝜃$ is a root of a monic polynomial $f(x)=x^{2^s}-32cp{x}^{2^r}-p\in \mathbb{Z}[x]$ with , then $\mathbb{Q}(𝜃)$ is not monogenic.