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Extra info for 1998 European Summer Meeting of the Association for Symbolic Logic. Logic Colloquium '98
In particular, K can have infinitely many 1998 ASSOCIATION FOR SYMBOLIC LOGIC EUROPEAN SUMMER MEETING 101 extensions of each degree. Using results of bounded arithmetic and some Galois theory we can prove the following results. THEOREM. Let M be a model of IAo a,, p E M a nonstandardprime and K the residue field. ),for n E N . Then there exists a unique abelian extension of K of dimension n. COROLLARY. For no prime r E N and k E N , K has two distinct normal extensions of degree rX. In the classical case every extension of a finite field is Galois and its Galois group is cyclic.
It. ) Let M be a model of IAo + and K be the residue field of M for a nonstandard prime pin M. e.. it satisfies the following axioms of Ax (1) there exists a unique extension of each degree n ; (2) every absolutely irreducible curve has a point in the field. In the case of Open Induction. Macintyre and Marker proved in  that for any field L of characteristic 0 there is a model M of Open Induction and a prime p E M such that the residue field K is elementary equivalent to L. In particular, K can have infinitely many 1998 ASSOCIATION FOR SYMBOLIC LOGIC EUROPEAN SUMMER MEETING 101 extensions of each degree.
The saturation axiom states that every class is p-saturated for an appropriate set p. This axiom implies the usual saturation for arbitrary classes of standard size which makes it possible to formalize the Loeb measure theory and a certain analog of the theorem on completeness of a nonstandard hull of a normed space. In many aspects NCT resembles Alternative Set Theory (AST) by P. Vopenka . First of all there are semisets (proper subclasses of sets) here and a theorem can be proved that a class is standard if and only if its intersection with any standard set is a standard set.