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Witt rings of infinite algebraic extensions of global fields. (English) Zbl 0923.11064

Let \(K\) be a field of characteristic not \(2\) such that its Witt ring \(WK\) is finitely generated. The “elementary type conjecture” then states that \(WK\) can be obtained from the Witt rings of the complex numbers, local or finite fields via direct products and group ring formations. It is shown in this paper that in the case where \(K\) is an algebraic extension of a global field (necessarily infinite in order that \(WK\) be finitely generated), then \(WK\) is in fact a direct product of Witt rings of local or finite fields.
One of the ingredients in the proof is an analogue of the Hasse-Minkowski local-global principle for infinite algebraic extensions of global fields which states that a quadratic form of rank \(\geq 3\) is isotropic over such an extension \(K\) if and only if it is isotropic over each localization of \(K\) (where localizations are defined in a suitable way). Clearly, this local-global principle also holds for forms of rank \(2\) if and only if every element in \(K\) which is a square in every localization is in fact a square in \(K\). The latter statement is true for global fields (“global square theorem”), but generally not for infinite algebraic extensions of global fields as is shown by an example.

MSC:

11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E08 Quadratic forms over local rings and fields
11E12 Quadratic forms over global rings and fields
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