Prime splitting and 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 $n$.
Fields of Study & Subjects
Advanced Algebra and Geometry
Algebraic Geometry and Number Theory
Analytic Number Theory Research
Publication details
- Journal / source
- Functiones et Approximatio Commentarii Mathematici
- Volume
- -1
- Issue
- -1
- Pages
- 1–30
- Publication date
- 1 January 2026