Rigid elements, valuations, and realization of Witt rings
Let T be a subgroup of the multiplicative group \(F^{\times}\) of a field F. Then \(a\in F^{\times}\) is said to be T-rigid, if \(T+aT\subseteq T\cup aT\). It turns out that T-rigid elements in F give rise to a valuation of the field F. This was first discovered by the third author [ibid. 68, 247-267 (1981; Zbl 0457.10007)] in case F is formally real and \(F^{\times 2}\subseteq T\) and then generalized to arbitrary fields by \textit{R. Ware} [Can. J. Math. 33, 1338-1355 (1981; Zbl 0514.10015)] still assuming \(F^{\times 2}\subseteq T.\) In the present paper these results are extended to cover the case where the squares do not necessarily sit in T. A section is devoted to adjusting the results to applications in quadratic form theory. Then the authors prove the following important realization theorem for Witt rings. If the Witt ring WF of a field F of characteristic not two is isomorphic to the product \(R\times S\) of two abstract Witt rings and if R is not basic (i.e. the abstract square class group \(G_ R\) contains birigid elements), then there is a 2-extension K of F such that the Witt ring WK is isomorphic to R. Moreover, the two isomorphisms, the natural morphism WF\(\to WK\) and the projection of \(R\times S\) onto R, combine into a commutative diagram. Application of the realization theorem in a forthcoming paper by the authors is announced. \{Another application to determining all profinite 2-groups which occur as the Galois groups of the maximal 2-extensions of fields with finite number of square classes can be found in a preprint by the third author and \textit{R. Ware}, ``Elementary Witt rings and pro-2 Galois groups'' (1987).\}
scientific article; zbMATH DE number 4023378
scientific article