Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of \(\mathbb{Q}\)
Let \(R\) be a subring of the field of algebraic numbers. The analogue of Hilbert's tenth problem for \(R\) is the question whether there is an algorithm which determines if an arbitrary given diophantine equation (with coefficients in \(R\)) has solutions in \(R\). For all known cases where \(R\) is a finitely generated \(\mathbb{Z}\)-module the problem has a negative answer and similar results are known in cases where \(R\) is a ring of ``\(S\)-integers'' of a number field (integers over a product of powers of a fixed finite set of primes). In contrast if \(R\) is the ring of all algebraic numbers the problem has a positive answer. The paper provides examples of rings \(R\) which are rings of \(S\)-integers of infinite extensions of \(\mathbb{Q}\) where Hilbert's tenth problem has a negative answer.
scientific article; zbMATH DE number 626269
scientific article