Pseudo algebraically closed field

Pseudo algebraically closed field

In mathematics, a field K is pseudo algebraically closed (usually abbreviated by PAC) if one of the following equivalent conditions holds:

*Each absolutely irreducible variety V defined over K has a K-rational point.
*Each absolutely irreducible polynomial fin K [T_1,T_2,cdots ,T_r,X] with frac{partial f}{partial X} ot =0 and for each nonzero gin K [T_1,T_2,cdots ,T_r] there exists ( extbf{a},b)in K^{r+1} such that f( extbf{a},b)=0 and g( extbf{a}) ot =0.
*Each absolutely irreducible polynomial fin K [T,X] has infinitely many K-rational points.
*If R is a finitely generated integral domain over K with quotient field which is regular over K, then there exist a homomorphism h:R o K such that h(a)=a for each ain K

Examples

* Algebraically closed fields and separably closed fields are always PAC.

* A non-principal ultraproduct of distinct finite fields is PAC.

* Infinite algebraic extensions of finite fields are PAC.

* This example arises from measure theory: The absolute Galois group G of a field K is profinite, hence compact, and hence equipped with a normalized Haar measure. Let K be a countable Hilbertian field and let e be a positive integer. Then for almost all e-tuple (sigma_1,...,sigma_e)in G^e, the fixed field of the subgroup generated by the automorphisms is PAC. Here the phrase "almost all" means "all but a set of measure zero".

References

* M. D. Fried and M. Jarden, Field Arithmetic, Second Edition, revised and enlarged by Moshe Jarden, Ergebnisse der Mathematik (3) 11, Springer, Heidelberg, 2004.


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