W-groups and values of binary forms
The \(W\)-group \(G_ F\) of a field \(F\) is the Galois group over \(F\) of the field \(F^{(3)}\) defined to be the compositum of \(F^{(2)}\) and of all quadratic extensions of \(F^{(2)}\) which are Galois extensions of \(F\), where \(F^{(2)}\) is the compositum of \(F\) and of all quadratic extensions of \(F\). The Witt ring of a field \(F\) determines the \(W\)-group up to isomorphism, and the \(W\)-group determines the Witt ring, unless every element of the field is a sum of two squares. The authors show that, more specifically, the value sets of binary quadratic forms over a field \(F\) can be described explicitly in terms of the maximal abelian quotient of the \(W\)-group. They also give criteria in terms of the \(W\)-group for a field \(F\) to have the level \(s(F)\) not greater than 4.
Galois groups of infinite field extensions
maximal abelian quotient of the \(W\)-group
scientific article; zbMATH DE number 219102