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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

##### MSC:
 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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##### References:
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