pseudo algebraically closed field

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.
• For each absolutely irreducible polynomial $f\backslash in\; K[T\_1,T\_2,\backslash cdots\; ,T\_r,X]$ with $\backslash frac\{\backslash partial\; f\}\{\backslash partial\; X\}\backslash not\; =0$ and for each nonzero $g\backslash in\; K[T\_1,T\_2,\backslash cdots\; ,T\_r]$ there exists $(\backslash textbf\{a\},b)\backslash in\; K^\{r+1\}$ such that $f(\backslash textbf\{a\},b)=0$ and $g(\backslash textbf\{a\})\backslash not\; =0$.
• Each absolutely irreducible polynomial $f\backslash in\; K[T,X]$ 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 $h:R\backslash to\; K$ such that $h(a)\; =\; a$ for each $a\; \backslash in\; K$.

• Algebraically closed fields and separably closed fields are always PAC.
• Pseudo-finite fields and hyper-finite fields are PAC.
• A non-principal ultraproduct of distinct finite fields is (pseudo-finite and henceFried & Jarden (2008) p.449) PAC.Fried & Jarden (2008) p.192 Ax deduces this from the Riemann hypothesis for curves over finite fields.
• Infinite algebraic extensions of finite fields are PAC.Fried & Jarden (2008) p.196
• 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 $e$-tuples $(\backslash sigma\_1,...,\backslash sigma\_e)\backslash in\; G^e$, 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".Fried & Jarden (2008) p.380 (This result is a consequence of Hilbert's irreducibility theorem.)
• Let K be the maximal totally real Galois extension of the rational numbers and i the square root of −1. Then K(i) is PAC.

• The Brauer group of a PAC field is trivial,Fried & Jarden (2008) p.209 as any Severi–Brauer variety has a rational point.
• The absolute Galois group of a PAC field is a projective profinite group; equivalently, it has cohomological dimension at most 1.Fried & Jarden (2008) p.210
• A PAC field of characteristic zero is C1.Fried & Jarden (2008) p.462

{{reflist}}

• {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd revised | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=Springer-Verlag | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 }}

Category:Algebraic geometry

Category:Field (mathematics)