Arithmetic over the ring of all algebraic integers. (English) Zbl 0581.14014

Rumely, Robert S.

J. Reine Angew. Math. 368, 127-133 (1986).

The main result of this paper is the proof of a conjecture of David Cantor, giving a Hasse principle for the ring of all algebraic integers: an absolutely irreducible affine variety defined over a number field K has points rational over the ring of all algebraic integers if and only if it has such points locally, for every finite place v of K. The proof is a generalization of a method due to D. C. Cantor and P. Roquette [J. Number Theory 18, 1-26 (1984; Zbl 0538.12014)], and is an application of capacity theory on algebraic curves as developed by the author. By combining the Hasse principle with constructive algebraic geometry and a decision procedure for completely valued algebraically closed fields, one obtains a positive solution to Hilbert’s tenth problem over the ring of algebraic integers: a decision procedure for diophantine equations and inequalities over this ring.

MSC:

14G05	Rational points
12L05	Decidability and field theory
14H25	Arithmetic ground fields for curves
11D75	Diophantine inequalities
11D99	Diophantine equations
03B25	Decidability of theories and sets of sentences

Keywords:

rational points of affine variety; Hasse principle; ring of all algebraic integers; capacity theory on algebraic curves; completely valued algebraically closed fields; Hilbert’s tenth problem; decision procedure for diophantine equations