Prime Splitting and Common $N$-Index Divisors in Radical Extensions: Part $p=2$

TL;DR

This paper explicitly describes the splitting of the prime 2 in radical extensions and classifies common index divisors, extending to common N-index divisors, with applications to non-monogenic fields and number rings.

Contribution

It provides a detailed analysis of prime splitting in radical extensions at p=2 and extends the classification of common index divisors to N-index divisors, including new constructions.

Findings

Explicit description of prime 2 splitting in radical extensions

Classification of common index divisors in these extensions

Construction of non-monogenic fields and rings with specific generator properties

Abstract

Following work of V\'elez, we explicitly describe the splitting of the integral prime 2 in the radical extension $\mathbb{Q}(\sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $\mathbb{Z}[x]$. With previous work of the second author, this fully describes the splitting of any prime in $\mathbb{Q}(\sqrt[n]{a})$. Using this description, we classify common index divisors (the primes whose splitting prevents the existence of a power integral basis for the ring of integers). Using work of Pleasants, we extend this to describe common $N$-index divisors (primes that divide the index of any order generated over $\mathbb{Z}$ by $N$ elements). We also present two novel constructions of non-monogenic fields with no common index divisors as well as constructions of number rings requiring $N$ ring generators for any $N>1$. Examples are provided throughout.