# 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 $fin K \left[T_1,T_2,cdots ,T_r,X\right]$ with $frac\left\{partial f\right\}\left\{partial X\right\} ot =0$ and for each nonzero $gin K \left[T_1,T_2,cdots ,T_r\right]$ there exists $\left( extbf\left\{a\right\},b\right)in K^\left\{r+1\right\}$ such that $f\left( extbf\left\{a\right\},b\right)=0$ and $g\left( extbf\left\{a\right\}\right) ot =0$.
*Each absolutely irreducible polynomial $fin K \left[T,X\right]$ 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 $h\left(a\right)=a$ 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 $\left(sigma_1,...,sigma_e\right)in G^e$, 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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