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On pseudo algebraically closed extensions of fields. (English) Zbl 1213.12006
The paper deals with the Pseudo Algebraically Closed (PAC) field extensions. Based on a generalization of the techniques used for embedding problems to field extensions, the paper proves a number of new results and gives alternative proofs to known results.
Among the new results, the following seem to be most relevant:
- Theorem 1, which establishes that the Galois closure of any proper separable algebraic PAC extension is its separable closure;
- Theorem 3, which gives a characterization of finite PAC extensions. More precisely, let \(K/K_0\) be a finite field extension. Then \(K/K_0\) is PAC if and only if one of the following holds:
(a) \(K_0\) is a PAC field and \(K/K_0\) is purely inseparable;
(b) \(K_0\) is real closed and \(K\) is its algebraic closure.

MSC:
12E30 Field arithmetic
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