Diophantine Generation, Galois Theory, and Hilbert's Tenth Problem

Abstract

Hilbert's Tenth Problem was a question concerning existence of an algorithm to determine if there were integer solutions to arbitrary polynomial equations over the integers. Building on the work by Martin Davis, Hilary Putnam, and Julia Robinson, in 1970 Yuri Matiyasevich showed that such an algorithm does not exist. One can ask a similar question about polynomial equations with coefficients and solutions in the rings of algebraic integers. In this thesis, we survey some recent developments concerning this extension of Hilbert's Tenth Problem. In particular we discuss how properties of Diophantine generation and Galois Theory combined with recent results of Bjorn Poonen, Barry Mazur, and Karl Rubin show that the Shafarevich-Tate conjecture implies that there is a negative answer to the extension of Hilbert's Tenth Problem to the rings of integers of number fields.