indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
Fields are finitely first-order axiomatizable. An algebraically closed field is further axiomatized by infinitely many sentences which state that all non-constant polynomials have a root. Once we additionally specify a characteristic , turns out to be complete, eliminates imaginaries, is stable, and admits quantifier elimination.
is the countable collection of sentences in the language of rings given by:
where .
We can additionally specify a characteristic to obtain by either adding the axioms to get or adding the axiom to get (where is prime).
has quantifier elimination. This amounts to a special case of Chevalley’s direct image theorem from algebraic geometry.
is stable.
is totally transcendental: Morley rank? is defined everywhere. In this setting Morley rank subsumes the usual Krull dimension of an algebraic variety.
is uncountably categorical: for each uncountable , models of of size must all be isomorphic. This fails at : has transcendence degree zero while there exist countable algebraically closed overfields of with infinite transcendence degree.
eliminates imaginaries. This means that its syntactic category has effective internal congruences, and has a good Galois theory.
It’s easy to see that codes all finite sets: if is a finite set of points inside a monster , its code is the tuple of coefficients for the polynomial
so that is fixed by an automorphism of if and only if that automorphism permutes .
Last revised on February 19, 2018 at 18:40:39. See the history of this page for a list of all contributions to it.