On the Kummer radical of $\mathbb{Z}_\ell$-extensions

TL;DR

This paper provides a new description of the Kummer radical for the first layers of $ ext{Z}_ ext{ell}$-extensions of number fields using inverse limits of norm maps, extending previous results and also describing the Tate kernel in $K_2(K)$.

Contribution

It introduces a novel approach to describe the Kummer radical in $ ext{Z}_ ext{ell}$-extensions via inverse limits, generalizing prior inclusions and connecting to Tate kernels.

Findings

New description of the Kummer radical using inverse limits.

Includes the inclusions established by Soogil Seo.

Provides a similar description for the Tate kernel in $K_2(K)$.

Abstract

On the basis of a previous work, we elaborate a new description of the Kummer radical associated to the first layers of $\mathbb{Z}_\ell$--extensions of a number fields K, by using inverse limits for the norm maps in the cyclotomic $\mathbb{Z}_\ell$-extension. Our main result contains, as an obvious consequence, the inclusions provided by Soogil Seo in a set of papers. By the same way we also give in the last section a similar description of the Tate kernel for universal symbols in $K_2(K)$.