Witt rings and almost free pro-2-groups
A field F is said to be quasi-pythagorean if the Kaplansky radical of F consists of all nonzero sums of two squares (or, equivalently, if \(I^ 2F\) is torsion free). It is proved here that, if F is a formally real quasi-pythagorean field and L/F is a maximal 2-extension such that \(\sum \dot F^ 2/\dot F^ 2\to\) \(\dot L^ 2/\dot L^ 2\) is injective, then the natural map WF \(\to\) \(W_{red}F\times WL\) is an isomorphism. Here WF, \(I^ 2F\) and \(W_{red}F\) are the Witt ring of F, the square of the fundamental ideal of WF and the reduced Witt ring of F. If moreover, F satisfies a finiteness condition on the space of orderings, then it follows that Gal(F(2)/F) is the free pro-2-product of Gal(K(2)/K) and \(Gal(F_{py}/F)\). Here F(2) is the quadratic closure of F and \(F_{py}\) is the Pythagorean closure of F. These results are used to give a new proof of a structure result for almost free pro-2-groups originally proved by \textit{Yu. L. Ershov} [Mat. Zametki 36, 913-923 (1984; Zbl 0573.12008)].
scientific article; zbMATH DE number 4158719
scientific article