# 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.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Field arithmetic — In mathematics, field arithmetic is a subject that studies the interrelations between arithmetic properties of a ql|field (mathematics)|field and its absolute Galois group.It is an interdisciplinary subject as it uses tools from algebraic number… …   Wikipedia

• Field (mathematics) — This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambiguation). In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it …   Wikipedia

• List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

• Absolute Galois group — In mathematics, the absolute Galois group GK of a field K is the Galois group of K sep over K , where K sep is a separable closure of K . Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K . The absolute… …   Wikipedia

• PAC — or Pac may refer to: Contents 1 People 2 Organizations and types of organizations 2.1 Political …   Wikipedia

• Outline of algebraic structures — In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety. Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither… …   Wikipedia

• List of algebraic structures — In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety. Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither… …   Wikipedia

• Model theory — This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. In mathematics, model theory is the study of (classes of) mathematical structures (e.g. groups, fields,… …   Wikipedia

• Schanuel's conjecture — In mathematics, specifically transcendence theory, Schanuel s conjecture is the following statement::Given any n complex numbers z 1,..., z n which are linearly independent over the rational numbers Q, the extension field Q( z 1,..., z n ,exp( z… …   Wikipedia

• Transcendence theory — In mathematics, transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.TranscendenceThe fundamental theorem of algebra tells us that if we have a non zero polynomial… …   Wikipedia