Geodesic
Local extensions associated with $l$-extensions with given ramification
Izvestiya. Mathematics , Tome 9 (1975) no. 4, pp. 693-726.

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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. 
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     author = {L. V. Kuz'min},
     title = {Local extensions associated with $l$-extensions with given ramification},
     journal = {Izvestiya. Mathematics },
     pages = {693--726},
     publisher = {mathdoc},
     volume = {9},
     number = {4},
     year = {1975},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1975_9_4_a1/}
}
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L. V. Kuz'min. Local extensions associated with $l$-extensions with given ramification. Izvestiya. Mathematics , Tome 9 (1975) no. 4, pp. 693-726. http://geodesic.mathdoc.fr/item/IM2_1975_9_4_a1/

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