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

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 a novel construction of non-monogenic fields with no common index divisors as well as constructions of number rings requiring $N+1$ ring generators for any $N>0$. Examples are provided throughout.