Pseudo algebraically closed field

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In mathematics, a field K 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 V defined over K has a K-rational point.
  • For each absolutely irreducible polynomial f\in K[T_1,T_2,\cdots ,T_r,X] with \frac{\partial f}{\partial X}\not =0 and for each nonzero g\in K[T_1,T_2,\cdots ,T_r] there exists (\textbf{a},b)\in K^{r+1} such that f(\textbf{a},b)=0 and g(\textbf{a})\not =0.
  • Each absolutely irreducible polynomial f\in K[T,X] has infinitely many K-rational points.
  • If R is a finitely generated integral domain over K with quotient field which is regular over K, then there exist a homomorphism h:R\to K such that h(a)=a for each a\in K

Examples[edit]

  • The PAC Nullstellensatz. The absolute Galois group G of a field K is profinite, hence compact, and hence equipped with a normalized Haar measure. Let K be a countable Hilbertian field and let e be a positive integer. Then for almost all e-tuple (\sigma_1,...,\sigma_e)\in G^e, the fixed field of the subgroup generated by the automorphisms is PAC. Here the phrase "almost all" means "all but a set of measure zero".[5] (This result is a consequence of Hilbert's irreducibility theorem.)

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