# Pseudo algebraically closed field

From Encyclopedia of Mathematics

*PAC field*

A field $K$ for which every non-empty algebraic variety defined over $K$ has a $K$-rational point. Clearly an algebraically closed field is PAC. The Brauer group of a PAC field is trivial.

More generally, let $O$ be a subset of $K$. The field $K$ is said to be PAC over $O$ if for every affine absolutely irreducible variety $V$ of dimension $n\ge0$ and for each dominating separable rational map $\phi : V \rightarrow A^r$ over $K$ there exists $a \in V(K)$ such that $\phi(a) \in O^r$. Each PAC field is PAC over itself.

See also Quasi-algebraically closed field.

#### References

- Fried, Michael D.; Jarden, Moshe
*Field arithmetic*(3rd revised ed.) Ergebnisse der Mathematik und ihrer Grenzgebiete. 3e Folge**11**Springer (2008) ISBN 978-3-540-77269-9 Zbl 1145.12001 - Jarden, Moshe; Razon, Aharon
*Pseudo algebraically closed fields over rings*Isr. J. Math.**86**(1994) 25-59 DOI 10.1007/BF02773673 Zbl 0802.12007

**How to Cite This Entry:**

Pseudo algebraically closed field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Pseudo_algebraically_closed_field&oldid=39974