Pseudo algebraically closed field

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In mathematics, a field is pseudo algebraically closed if it satisfies certain properties which hold for any algebraically closed field. 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