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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2010, Volume 10, Issue 2, Pages 29–50 (Mi adm47)

RESEARCH ARTICLE

2-Galois groups and the Kaplansky radical

R. P. Darioa, A. Englerb

a UTFPR-DAMAT, Av. Sete de Setembro, 3165, 80230-901 Curitiba, PR, Brasil
b UNICAMP-IMECC, Caixa Postal 6065, 13083-970 Campinas, SP, Brasil

Abstract: 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, Galois group, 2-henselian valuation ring, quadratic form.

Full text: PDF file (316 kB)

Bibliographic databases:
MSC: 12J10, 12F10
Revised: 01.03.2011
Language:

Citation: R. P. Dario, A. Engler, “2-Galois groups and the Kaplansky radical”, Algebra Discrete Math., 10:2 (2010), 29–50

Citation in format AMSBIB
\Bibitem{DarEng10}
\by R.~P.~Dario, A.~Engler
\paper 2-Galois groups and the Kaplansky radical
\jour Algebra Discrete Math.
\yr 2010
\vol 10
\issue 2
\pages 29--50
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2884742}
\zmath{https://zbmath.org/?q=an:1224.12006}