Difference between revisions of "Pseudo algebraically closed field"
From Encyclopedia of Mathematics
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(PAC over a ring, cite Jarden and Razon) |
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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. | 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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| + | 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]]. | See also [[Quasi-algebraically closed field]]. | ||
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====References==== | ====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}} | * 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}} | ||
Revision as of 12:44, 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=37154
Pseudo algebraically closed field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo_algebraically_closed_field&oldid=37154