Geodesic
2-Galois groups and the Kaplansky radical
Algebra and discrete mathematics, Tome 10 (2010) no. 2, pp. 29-50

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

An accurate description of the Galois group $G_{F}(2)$ of the maximal Galois 2-extension of a field $F$ may be given for fields $F$ admitting a 2-henselian valuation ring. In this note we generalize this result by characterizing the fields for which $G_{F}(2)$ decomposes as a free pro-2 product $\mathcal{F}*\mathcal{H}$ where $\mathcal{F}$ is a free closed subgroup of $G_{F}(2)$ and $\mathcal{H}$ is the Galois group of a 2-henselian extension of $F$. The free product decomposition of $G_{F}(2)$ is equivalent to the existence of a valuation ring compatible with the Kaplansky radical of $F$. Fields with Kaplansky radical fulfilling prescribed conditions are constructed, as an application.
Keywords: Brauer group, free pro-2 product, 2-henselian valuation ring, quadratic form.
Mots-clés : Galois group 
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     author = {R. P. Dario and A. Engler},
     title = {2-Galois groups and the {Kaplansky} radical},
     journal = {Algebra and discrete mathematics},
     pages = {29--50},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2010_10_2_a3/}
}
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R. P. Dario; A. Engler. 2-Galois groups and the Kaplansky radical. Algebra and discrete mathematics, Tome 10 (2010) no. 2, pp. 29-50. http://geodesic.mathdoc.fr/item/ADM_2010_10_2_a3/