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 $f\in K[T_1,T_2,\cdots ,T_r,X]$ with $\frac{\partial f}{\partial X}\not =0$ and for each nonzero $g\in K[T_1,T_2,\cdots ,T_r]$ there exists $(\textbf{a},b)\in K^{r+1}$ such that $f(\textbf{a},b)=0$ and $g(\textbf{a})\not =0$ .
Each absolutely irreducible polynomial $f\in K[T,X]$ 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 $h:R\to K$ such that $h (a) = a$ for each $a\in K$

[edit] Examples

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".

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