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

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 $K_2(K)$.

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