A universal first-order formula defining the ring of integers in a number field

We show that the complement of the ring of integers in a number field $K$ is Diophantine, for $f \in K[t, x_1, \dots , x_n]$. We will use global class field theory and generalize the ideas originating from Koenigsmann’s recent result giving a universal first-order formula for $\mathbb{Z}$ in $\mathbb{Q}$.

Keywords

Hilbert’s tenth problem, Diophantine set, quaternion algebra, class field theory, Artin reciprocity, Hilbert symbol

2010 Mathematics Subject Classification

Primary 11R37. Secondary 11R52, 11U05.

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Published 28 April 2014