Pseudo algebraically closed field

In mathematics, a field $K$ is pseudo algebraically closed if it satisfies certain properties which hold for any algebraically closed field. The concept was introduced by James Ax in 1967.[1]

Formulation

A field K is pseudo algebraically closed (usually abbreviated by PAC[2]) if one of the following equivalent conditions holds:

• Each absolutely irreducible variety $V$ defined over $K$ has a $K$-rational point.
• For 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 $a\in K$

Examples

• The PAC Nullstellensatz. 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".[5] (This result is a consequence of Hilbert's irreducibility theorem.)

References

1. ^ a b Fried & Jarden (2008) p.218
2. ^ a b Fried & Jarden (2008) p.192
3. ^ Fried & Jarden (2008) p.449
4. ^ Fried & Jarden (2008) p.196
5. ^ Fried & Jarden (2008) p.380
6. ^ Fried & Jarden (2008) p.209
7. ^ a b Fried & Jarden (2008) p.210
8. ^ Fried & Jarden (2008) p.462