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On PAC extensions and scaled trace forms. (English) Zbl 1251.11020
Let \(K\) be a field of characteristic \(\neq 2\).
For each finite separable field extension \(L/K\), any nonzero element \(\alpha \in L\) gives rise to a non-degenerate quadratic form \(L \rightarrow K\) defined by \(x\mapsto \text{Tr}_{F/K}(\alpha x^2)\). These forms are called scaled trace forms.
This paper deals with the problem of determining which non-degenerate quadratic forms over \(K\) are isomorphic to scaled trace forms. The main result (Thm. 4) states that, if there exists a PAC (pseudo algebraically closed) extension \(M/K\) such that \(M\) admits a separable extension of degree \(n\), then every non-degenerate quadratic form of dimension \(n\) over \(K\) is isomorphic to a scaled trace form.
From this, the authors deduce that the same conclusion holds in each of the following cases: (1) \(K\) is a prosolvable extension of a Hilbertian field, and \(n>4\); (2) \(K\) is a separable extension of a Hilbertian field \(K_0\) such that, for some prime \(p\), the Galois closure of \(K/K_0\) is prime-to-\(p\) over \(K_0\), and \(n\) is an arbitrary positive integer.
The proof of the main result is inspired by the arguments used by W. Scharlau [Math. Z. 196, 125–127 (1987; Zbl 0658.10025)] and W. C. Waterhouse [Arch. Math. 47, 229–231 (1986; Zbl 0607.10013)] to show that, if \(K\) is Hilbertian, then every non-degenerate quadratic form over \(K\) is isomorphic to a scaled trace form.

MSC:
11E04 Quadratic forms over general fields
12E30 Field arithmetic
12E25 Hilbertian fields; Hilbert’s irreducibility theorem
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