  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 $K_2(K)$.
