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Difference between revisions of "Pseudo algebraically closed field"

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(PAC over a ring, cite Jarden and Razon)
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''PAC field''
 
''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.
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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.
 
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.

Revision as of 19:14, 11 December 2016


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