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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:
 1.2e+31 Field arithmetic
##### Keywords:
field arithmetic; PAC; embedding problem
Full Text:
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