Jump to content

Pseudo algebraically closed field

From Wikipedia, the free encyclopedia

In mathematics, a field is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.[1]

Formulation

[edit]

A field K is pseudo algebraically closed (usually abbreviated by PAC[2]) if one of the following equivalent conditions holds:

  • Each absolutely irreducible variety defined over has a -rational point.
  • For each absolutely irreducible polynomial with and for each nonzero there exists such that and .
  • Each absolutely irreducible polynomial has infinitely many -rational points.
  • If is a finitely generated integral domain over with quotient field which is regular over , then there exist a homomorphism such that for each .

Examples

[edit]

Properties

[edit]

References

[edit]
  1. ^ a b Fried & Jarden (2008) p.218
  2. ^ a b Fried & Jarden (2008) p.192
  3. ^ Fried & Jarden (2008) p.449
  4. ^ Fried & Jarden (2008) p.196
  5. ^ Fried & Jarden (2008) p.380
  6. ^ Fried & Jarden (2008) p.209
  7. ^ a b Fried & Jarden (2008) p.210
  8. ^ Fried & Jarden (2008) p.462
  • Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.