Pseudo algebraically closed field

From Encyclopedia of Mathematics
Jump to: navigation, search

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.


  • 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: