Geodesic
Local extensions associated with $l$-extensions with given ramification
Izvestiya. Mathematics, Tome 9 (1975) no. 4, pp. 693-726 
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/
@article{IM2_1975_9_4_a1,
     author = {L. V. Kuz'min},
     title = {Local extensions associated with $l$-extensions with given ramification},
     journal = {Izvestiya. Mathematics},
     pages = {693--726},
     year = {1975},
     volume = {9},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1975_9_4_a1/}
}
TY  - JOUR
AU  - L. V. Kuz'min
TI  - Local extensions associated with $l$-extensions with given ramification
JO  - Izvestiya. Mathematics
PY  - 1975
SP  - 693
EP  - 726
VL  - 9
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/IM2_1975_9_4_a1/
LA  - en
ID  - IM2_1975_9_4_a1
ER  - 
%0 Journal Article
%A L. V. Kuz'min
%T Local extensions associated with $l$-extensions with given ramification
%J Izvestiya. Mathematics
%D 1975
%P 693-726
%V 9
%N 4
%U http://geodesic.mathdoc.fr/item/IM2_1975_9_4_a1/
%G en
%F IM2_1975_9_4_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Kuzmin L. V., “Gomologii prokonechnykh grupp, multiplikator Shura i teoriya polei klassov”, Izv. AN SSSR. Ser. matem., 33 (1969), 1220–1254 | MR

[2] Kuzmin L. V., “Modul Teita polei algebraicheskikh chisel”, Izv. AN SSSR. Ser. matem., 36 (1972), 267–327 | MR

[3] Kokh X., Teoriya Galua $p$-rasshirenii, Mir, M., 1973 | MR

[4] Brumer Z., “Galois groups of extensions of algebraic number fields with given ramification”, Mich. Math. J., 13:1 (1966), 33–40 | DOI | MR | Zbl

[5] Kuzmin L. V., “Kogomologicheskaya razmernost nekotorykh grupp Galua”, Izv. AN SSSR. Ser. matem., 39:3 (1975), 487–495 | MR

[6] Borevich Z. I., Shafarevich I. R., Teoriya chisel, Nauka, M., 1964 | MR

[7] Neukirch J., “Einbettungsprobleme mit lokaler Vorgabe und freie Producte lokaler Galoisgruppen”, J. Reine und Angew. Math., 259 (1973), 1–47 | MR | Zbl