A Diophantine definition of rational integers over some rings of algebraic numbers
After a negative answer was given to Hilbert's Tenth Problem (that is, is there an algorithm that identifies the diophantine equations which have rational integer solutions and those that don't?) it is natural to ask a similar question in various other domains, for example, in the ring of integers \({\mathcal O}_ K\) of a number field \(K\). In all known cases (notably for \(K\) a quadratic field, a totally real extension or an abelian extension) the problem has a negative answer. Yet, the problem remains open for an arbitrary number field \(K\). The author gives a negative answer to the similar question for domains \({\mathcal O}\) which result from an \({\mathcal O}_ K\) by inverting a finite set of primes, for each number field \(K\) where the problem has been solved. The approach is in the classical style (first developed by Denef- Lipshitz): one considers the solutions of a Pell equation in \({\mathcal O}\). These correspond to a group of units with finite rank in a quadratic extension of \(K\). Then one essentially defines in a positive-existential way a subgroup of rank one (this is the hard part) and uses this in order to give a positive-existential definition of the rational integers in \({\mathcal O}\). The final result follows by the answer to Hilbert's Tenth Problem.
diophantine definition of rational integers
scientific article; zbMATH DE number 120367
scientific article