Hereditarily quadratically closed fields
The authors prove for a finitely generated field extension K/F with \(tr.\deg_ FK=n\geq 1\) and char(F)\(\neq 2\) the equivalence of the following statements: (1) F(\(\sqrt{-1})\) and all its algebraic extensions are quadratically closed; (2) \(cd_ 2K(\sqrt{-1})=n\), where \(cd_ 2(F)\) means the cohomological 2-dimension of the absolute Galois group of the field F; (3) \(u(K(-1))=2^ n\) for \(u(F)=:\sup \{\dim (q)| \quad q\) is an anisotropic F-quadratic form with zero signature in each real closure of \(F\}\) ; (4) \(I^{n+1}K\) ( \(= the\) ideal in the Witt ring of K generated by the n-fold Pfister forms, i.e. forms of type \(\otimes^{n}_{i=1}<1,a_ i> )\) contains no \(n+1\)-fold anisotropic Pfister form having zero signature in each real closure of K; (5) \(I^{n+1}K\) is torsion-free; (6) min\(\{\) \(r|\) \(I^{r+1}K=2I^ rK\} \leq n\).
scientific article; zbMATH DE number 4095624
scientific article