Primes and their residue rings in models of open induction
This paper deals with the class OI of models for open induction \((+\) the axioms for ordered rings) especially with ``pathological'' algebraic properties of such models and corresponding independence results related mainly to the ordinary arithmical properties of prime numbers. [Cf. \textit{J. C. Shepherdson}, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 12, 79-86 (1964; Zbl 0132.247), \textit{A. J. Wilkie}, Logic Colloquium '77, Stud. Logic Found. Math. 96, 285-296 (1978; Zbl 0449.03076), \textit{L. P. D. van den Dries}, Lect. Notes Math. 834, 346-362 (1980; Zbl 0454.03034).] It has been known that OI does not prove the cofinality of the set of primes. Here the authors prove, e.g., that the following may happen in a ring \(R\in OI:\) (A) In general, \(\alpha\in R\) is an irreducible (i.e., has no proper factorization) \(\nrightarrow\) \(\alpha\) is a prime (i.e., R/\(\alpha\) R is a domain); also, \(\alpha\) is a prime \(\nrightarrow\) R/\(\alpha\) R is a field. (B) The order type of the infinite primes may be arbitrary. (C) All the infinite primes may be congruent to 1 mod 4 (resp. 3 mod 4). (D) The residue field of an infinite prime may be any characteristic 0 field of infinite transcendence degree. (E) The field of fractions may be elemenarily equivalent to any ordered field which is dense in its real closure. (F) Schinzel's hypothesis (which is a generalization of the hypothesis that there are infinitely many pairs of twin primes) may hold. (G) Every infinite even integer may be the sum of two primes. Here, except case (E), R is normal (i.e., integrally closed in F(R)). Note that from (D) we see, e.g., that OI does not suffice to give quadratic non-residues for odd primes. Indeed, the residue field of such a prime may be algebraically closed. In the case of a real closed residue field such a prime cannot be a sum of squares, so Lagrange's theorem fails here. Moreover, there is a normal model of OI in which the primes are cofinal and no infinite prime is a sum of squares. Also, there is a (nonnormal) model R of OI in which Lagrange's theorem fails, but every positive element is a square of an element of F(R). All these models are constructed by union of chains arguments, involving repeated use of certain purely algebraic constructions. There are also some open problems in the paper. E.g.: is there a recursive nonstandard model of OI with an infinite prime?
scientific article; zbMATH DE number 4104953