Primes and their residue rings in models of open induction
type: Item
instance of: scholarly article
title: Primes and their residue rings in models of open induction
zbMATH Open document ID: 0675.03040
DOI: 10.1016/0168-0072(89)90025-0
author: David Marker
author: Angus Macintyre
published in: Annals of Pure and Applied Logic
publication date: 1989-01-01T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 03H15
Mathematics Subject Classification ID: 03C62
Mathematics Subject Classification ID: 11U10
Mathematics Subject Classification ID: 12L15
zbMATH DE Number: 4104953
zbMATH Keywords: models for open induction
zbMATH Keywords: ordered rings
zbMATH Keywords: independence results
zbMATH Keywords: prime numbers
zbMATH Keywords: infinite primes
zbMATH Keywords: residue field
zbMATH Keywords: ordered field
reviewed by: Vladimir Yu. Sazonov
MaRDI profile type: Publication
full work available at URL: 0168-0072(89)90025-0
OpenAlex ID: W1965416269
cites work: Open induction and the true theory of rationals
cites work: A Linearly Ordered Ring whose Theory Admits Elimination of Quantifiers is a Real Closed Field
cites work: Q5611879
cites work: On the scheme of induction for bounded arithmetic formulas
cites work: Q5341750
cites work: Q3895478
description: scientific article; zbMATH DE number 4104953