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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:
 110000 Quadratic forms over general fields 1.2e+31 Field arithmetic 1.2e+26 Hilbertian fields; Hilbert’s irreducibility theorem
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