The unit norm index and $p$-class group in certain degree $\ell$ extensions

TL;DR

This paper investigates the conditions under which units in a number field are norms from an extension, and applies these findings to analyze the structure of $ ext{ extit{ extltilde}}$-class groups in specific $ ext{ extltilde}$-extensions of $ ext{ extltilde}$-rational fields.

Contribution

It introduces new criteria for units to be norms in certain extensions and applies these to study $ ext{ extltilde}$-class groups in anti-cyclotomic $ ext{ extltilde}$-extensions.

Findings

Established conditions for units to be norms in degree $ ext{ extltilde}$ extensions.

Analyzed the $ ext{ extltilde}$-class group structure in lifts of anti-cyclotomic $ ext{ extltilde}$-extensions.

Provided insights into the behavior of $ ext{ extltilde}$-class groups in specific number field extensions.

Abstract

We examine when units in a field are the norms of elements in an extension field, given certain conditions. We apply these results to the study of the $\ell$-class groups in lifts of the anti-cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(i)$.