Discriminant and integral basis of pure nonic fields

TL;DR

This paper determines the prime factorization of the index of the subring generated by a root of x^9 - a in the ring of integers of pure nonic fields, providing explicit p-integral bases and constructing an integral basis.

Contribution

It offers the exact prime powers dividing the index and constructs p-integral bases for pure nonic fields, leading to explicit integral bases with examples.

Findings

Exact prime power divisors of the index are identified.

Explicit p-integral bases are provided for each prime.

A method to construct an integral basis from p-integral bases is demonstrated.

Abstract

Let $K = \Q(\theta)$ be an algebraic number field with $\theta$ satisfying an irreducible polynomial $x^{9} - a$ over the field $\Q$ of rationals and $\Z_K$ denote the ring of algebraic integers of $K$. In this article, we provide the exact power of each prime which divides the index of the subgroup $\Z[\theta]$ in $\Z_K$. Further, we give a $p$-integral basis of $K$ for each prime $p$. These $p$-integral bases lead to a construction of an integral basis of $K$ which is illustrated with examples.