Principalization of ideals in abelian extensions of number fields

TL;DR

This paper proves a conjecture that in certain abelian extensions of number fields, all ideals become principal in the compositum of the extension with the maximal abelian extension of the base field, confirming a long-standing hypothesis.

Contribution

It provides a complete proof of Georges Gras's conjecture regarding ideal principalization in specific abelian extensions of number fields.

Findings

All ideals in the extension become principal in the compositum with the maximal abelian extension.

The proof confirms the conjecture for extensions with at least one totally split infinite place.

The result advances understanding of ideal class behavior in abelian extensions.

Abstract

We give the complete proof of a conjecture of Georges Gras which claims that, for any extension $K/k$ of number fields in which at least one infinite place is totally split, every ideal $I$ of $K$ principalizes in the compositum $Kk^{ab}$ of $K$ with the maximal abelian extension $k^{ab}$ of $k$