Prime Splitting and Common Index Divisors in Radical Extensions

TL;DR

This paper explicitly describes how odd primes split in radical extensions and investigates primes that obstruct the existence of power integral bases, showing such primes divide the extension degree n.

Contribution

It provides explicit prime splitting descriptions in radical extensions and characterizes primes that serve as common index divisors.

Findings

Odd primes split explicitly in radical extensions.

Primes obstructing power integral bases divide n.

Characterization of common index divisors in radical extensions.

Abstract

We explicitly describe the splitting of odd integral primes in the radical extension $\mathbb{Q}(\sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $\mathbb{Z}[x]$. Our motivation is to classify common index divisors, the primes whose splitting provides a local obstruction to the existence of a power integral basis for the ring of integers of $\mathbb{Q}(\sqrt[n]{a})$. Among other results, we show that if $p$ is such a prime, even or otherwise, then $p$ divides $n$.