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On trace forms of algebraic number fields. (English) Zbl 0777.11009
Let $$K$$ be a field of characteristic $$\neq 2$$, $$A/K$$ a finite dimensional commutative and étale algebra over $$K$$. The quadratic form $$A\to K: x\mapsto \text{tr}_{A/K}(x^ 2)$$ is called trace form. Let $$\psi$$ be a nondegenerate quadratic form of dimension $$n\geq 4$$ over $$K$$. We prove the following. If $$K$$ is an algebraic number field, then there exists a field extension $$L/K$$ such that the trace form of $$L/K$$ is isometric to $$\psi$$ if and only if no signature of $$\psi$$ is negative. Each trace form of an étale algebra of odd dimension $$\geq 3$$ defined over a Hilbertian field $$K$$ of characteristic 0 is a trace form of a field extension.

##### MSC:
 1.1e+13 Quadratic forms over global rings and fields 110000 Quadratic forms over general fields 1.1e+82 Algebraic theory of quadratic forms; Witt groups and rings
##### Keywords:
étale algebra; quadratic form; trace form; field extension
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##### References:
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