Pseudo algebraically closed field
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A field K is pseudo algebraically closed (usually abbreviated by PAC) 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
- Algebraically closed fields and separably closed fields are always PAC.
- Pseudo-finite fields and hyper-finite fields are PAC.
- A non-principal ultraproduct of distinct finite fields is (pseudo-finite and hence) PAC. Ax deduces this from the Riemann hypothesis for curves over finite fields.
- Infinite algebraic extensions of finite fields are PAC.
- The PAC Nullstellensatz. The absolute Galois group of a field is profinite, hence compact, and hence equipped with a normalized Haar measure. Let be a countable Hilbertian field and let be a positive integer. Then for almost all -tuple , 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". (This result is a consequence of Hilbert's irreducibility theorem.)
- Let K be the maximal totally real Galois extension of the rational numbers and i the square root of -1. Then K(i) is PAC.
- The Brauer group of a PAC field is trivial, as any Severi–Brauer variety has a rational point.
- The absolute Galois group of a PAC field is a projective profinite group; equivalently, it has cohomological dimension at most 1.
- A PAC field of characteristic zero is C1.
- Fried & Jarden (2008) p.218
- Fried & Jarden (2008) p.192
- Fried & Jarden (2008) p.449
- Fried & Jarden (2008) p.196
- Fried & Jarden (2008) p.380
- Fried & Jarden (2008) p.209
- Fried & Jarden (2008) p.210
- Fried & Jarden (2008) p.462