Quadratic forms and Galois cohomology over formally real fields
Let \(F\) be a non-Pythagorean field of characteristic different from two and let the third power \(I^ 3 F\) of the fundamental ideal \(IF\) of the Witt ring of \(F\) be torsion free. The main result of the paper is as follows. If \(I^ 3 F(\sqrt{-1})=0\), or, if every binary torsion form represents a totally positive element of the field \(F\), then there exists an exact sequence \(1\to{\mathcal F}\to G_ F\pi\to\mathbb{Z}_ 2\to 1\) where \({\mathcal F}\) is a free pro-2-group and \(G_ F \pi\) is the Galois group of the Pythagorean closure of \(F\) over \(F\). This generalizes earlier results of the author [J. Pure Appl. Algebra 30, 95-107 (1983; Zbl 0518.10024)] valid only for algebraic number fields. As corollaries the author obtains complete characterizations in terms of Galois groups of fields \(F\) with \(I^ 3 F=0\), or \(I^ 2 F\) torsion free, or, under additional hypotheses, of fields with \(I^ 3 F\) torsion free.
Galois groups of infinite field extensions
scientific article; zbMATH DE number 218571