The Steinitz Realization Problem

TL;DR

This paper proves that every element of the ideal class group of a number field can be realized as the Steinitz class of some degree n extension, solving the Steinitz realization problem completely.

Contribution

It provides a complete solution to the Steinitz realization problem for all degrees and number fields, establishing that all ideal class group elements are realizable as Steinitz classes.

Findings

All ideal class group elements are realizable as Steinitz classes.

The result holds for all degrees n and all number fields K.

The problem is affirmatively solved in general.

Abstract

Let $K$ be a number field and let $n \in \mathbb{Z}_{>1}$. The Steinitz realization problem asks: does every element of the ideal class group of $K$ occur as the Steinitz class of a degree $n$ extension of $K$? In this article, we give an affirmative answer to the Steinitz realization problem for all $n$ and $K$.