Abstract

The aim of this work is to retrace the path to the solution of the classical Hilbert's Tenth Problem and the attempts trying to generalize it reaching, finally, a new result concerning its extension in number fields. First we deal with the classical case classifying diophantine functions as the recursive ones, using the diophantinity of the exponential as a key tool. In order to generalize the unsolvability of the diophantine problem to rings of integers of number fields the line is to simulate the core of the classicl proof, which seems to be more natural in the setting of totally real number fields as shown by the work of Denef and Lipshitz. After that few attempts were done to reach a general result, Pheidas was able to use the Elliptic Curves to obtain a new criteria which, in its generalized version of Shlapentokh, is the key together with the new additive combinatorics techniques of Pagano-Koymans to the proof of the general case.