View source for Pseudo algebraically closed field

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''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}}