On relative pure cyclic fields with power integral bases

Abstract

Let $L = K(\alpha)$ be an extension of a number field $K$, where $\alpha$ satisfies the monic irreducible polynomial $P(X)=X^p-\beta$ of prime degree belonging to $\mathfrak{o}_K[X]$ ($\mathfrak{o}_K$ is the ring of integers of $K$). The purpose of this paper is to study the monogenity of $L$ over $K$ by a simple and practical version of Dedekind's criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field $L$ with a pure cubic subfield, which is not necessarily a composite extension of two cubic subfields. We obtain a slightly simpler computation of the discriminant $d_{L/\mathbb{Q}}$.

Keywords

• discrete valuation ring
• dedekind ring
• monogenity
• relative integral basis
• nonic field