Constructing quasi-pythagorean fields
A field \(F\) is said to be quasi-pythagorean if every sum of squares in \(F\) is a norm from every quadratic extension of \(F\). Equivalently, \(u(F)\leq 2\), and also, \(I^ 2F\) is torsion free. \textit{R. Elman} and \textit{T. Y. Lam} [Math. Ann. 219, 21-42 (1976; Zbl 0302.10025)] characterized the fields as satisfying the property \(A_ 2\). In an earlier paper the author [Wiss. Beitr., Martin-Luther-Univ. Halle- Wittenberg 1987/33 (M48), 175-182 (1987; Zbl 0628.12015)] found a criterion for quasi-pythagoreanity of an algebraic extension of the rational number field \(\mathbb{Q}\). The criterion was shown to imply the quasi-pythagoreanity of the field \(F\) obtained from \(\mathbb{Q}\) by adjoining all two-power roots of all prime numbers. In the present paper the author shows that adjoining to a global field all two-power roots of finitely many elements \(a_ 1,\dots,a_ r\) of the field, does not produce a quasi-pythagorean field unless \(-1\) is in the multiplicative group generated by all the two-power roots of \(a_ 1,\dots,a_ r\).
scientific article; zbMATH DE number 95927