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.