Diophantine Sets Over Algebraic Integer Rings. II

Abstract

We prove that Z is diophantine over the ring of algebraic integers in any totally real number field or quadratic extension of a totally real number field. 1. Introduction.2Let B be a commutative ring with unit and let R(xx, . . ., xn) be a relation in B (in the sense of set theory).We say that R(xx, . . ., xn) is diophantine over B if there exists a polynomial P(xx, . . ., x",yx, . . .,ym) with coefficients in B such that, for all x,, ... , x" in B, R(xx,. ..,x")<H>3yx, ...,ymG B: P(xx, . . ., xn,yx, . . .,ym) = 0