Local extensions associated with $l$-extensions with given ramification

L. V. Kuz'min

Izvestiya. Mathematics , Tome 9 (1975) no. 4, pp. 693-726

Résumé

Let $l$ be a prime number, $k$ an algebraic number field containing a primitive $l$th root of unity, $S$ a finite set of valuations of $k$ containing all prime divisors of $l$, and $K$ the maximal $l$-extension of $k$ unramified outside $S$. The paper studies local extensions $K_v/k_v$ for $v\in S$, and the corresponding decomposition subgroups $G_v\subset G(K/k)$. It is proved that in almost all cases $K$ coincides with the maximal $l$-extension of $k$; in particular, this holds if $G_v\ne G(K/k)$. Also, a series of results is obtained on the relative location of the various $G_v$ in $G$, and the group of universal norms from the group of $S$-units of $K$ to the group of $S$-units of $k$ is computed. Bibliography: 7 items.