Let f ϵ Z [ X, Y ] be irreducible. We give a condition that there are only finitely many integers n ϵ Z such that f is reducible and we give a bound for such integers. We prove a similar result for polynomials with coefficients in polynomial rings. Both results are proved by, so-called, nonstandard arithmetic.

LetKbe an algebraic number field andIKthe ring of algebraic integers inK. *Kand *IKdenote enlargements ofKandIKrespectively. LetxЄ *K–K. In this paper, we are concerned with algebraic extensions ofKwithin *K. For eachxЄ *K–Kand each natural numberd, YKis defined to be the number of algebraic extensions ofKof degreedwithin *K.xЄ *K–Kis called a Hilbertian element ifYK= 0 for alldЄ N,d> 1; in other words,Khas no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a (...) Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural numberω, 2ωPωand 2ω are Hilbertian elements in*Q, where pωis theωth prime number. (shrink)

We prove that PTCN cannot be a model of U12. This implies that there exists a first order sentence of bounded arithmetic which is provable in U12 but does not hold in PTCN.