Characterisation of Primes Dividing the Index of a Class of Polynomials and Its Applications

Abstract

Let ${\mathbb {Z}}_{K}$ denote the ring of algebraic integers of an algebraic number field $K = {\mathbb Q}(\theta )$ , where $\theta $ is a root of a monic irreducible polynomial $f(x) = x^n + a(bx+c)^m \in {\mathbb {Z}}[x]$ , $1\leq m<n$ . We say $f(x)$ is monogenic if $\{1, \theta , \ldots , \theta ^{n-1}\}$ is a basis for ${\mathbb {Z}}_K$ . We give necessary and sufficient conditions involving only $a, b, c, m, n$ for $f(x)$ to be monogenic. Moreover, we characterise all the primes dividing the index of the subgroup ${\mathbb {Z}}[\theta ]$ in ${\mathbb {Z}}_K$ . As an application, we also provide a class of monogenic polynomials having non square-free discriminant and Galois group $S_n$ , the symmetric group on n letters.