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On trace forms of algebraic number fields. (English) Zbl 0658.10025
Let L/K be a finite separable field extension and let \(Tr_{L/K}<1>\) denote the associated trace form on \(L\times L\) over K given by \((x,y)\mapsto Tr_{L/K}(xy)\). For any ordering P of K the signature \(sign_ P(Tr_{L/K}<1>)\) is nonnegative. In the paper under review it is shown that for algebraic number fields K the converse is true: Theorem: Let K be an algebraic number field with Witt ring W(K). For any \(\phi\in W(K)\) assume \(sign_ P(\phi)\geq 0\) for all orderings P of K. Then there exists a finite extension L/K such that \(\phi\) is Witt equivalent to \(Tr_{L/K}<1>.\)
This generalizes similar results obtained by Conner and Perlis in the case \(K={\mathbb{Q}}\). The proof uses a certain version of Hilbert’s irreducibility theorem.
Reviewer: H.-J.Bartels

11E16 General binary quadratic forms
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
11E12 Quadratic forms over global rings and fields
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