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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:
11E12 Quadratic forms over global rings and fields
11E04 Quadratic forms over general fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
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