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Constructing quasi-pythagorean fields. (English) Zbl 0767.11020

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. R. Elman and 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 \(a_ 1,\dots,a_ r\).

MSC:

11E10 Forms over real fields
11E12 Quadratic forms over global rings and fields
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
11E81 Algebraic theory of quadratic forms; Witt groups and rings
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