# Meeting Details

Title: Hilbert's Tenth Problem in complementary subrings of number fields Algebra and Number Theory Seminar Kirsten Eisentraeger, Penn State University In this talk I will prove that Hilbert's Tenth Problem is undecidable for subrings of algebraic number fields that are complementary in a very strong sense. I will show that the non-archimedean primes of a number field $K$ can be partitioned into $t$ disjoint recursive subsets $S_1, \dots, S_t$ (for any $t >1$) such that Hilbert's Tenth Problem is undecidable for the corresponding rings of $S_i$-integers. The only assumption on $K$ is that there is an elliptic curve of rank one defined over $K$. This is joint work with Graham Everest and Alexandra Shlapentokh.