Generalization of Fermat’s last theorem to real quadratic fields

Abstract

The main purpose of this master thesis is to study a generalization of Fermat’s Last Theorem for real quadratic fields. As it is well-known, Fermat’s Last Theorem states that the equation $a^{n}+b^{n}=c^{n}, abc \not={0}$ has no integer solutions when the exponent $n$ is greater or equal than 3. It was enunciated by Fermat around 1630 and stood unsolved for more than 350 years, until 1994 Andrew Wiles finally took that last step by proving the modularity conjecture for semistable elliptic curves. The whole proof of FLT involves mathematical tools which are widely used in Number Theory. Namely, elliptic curves, modular forms and Galois representations. It entangles contributions by many authors, for instance; the work of Frey, who attached an elliptic curve with some ”remarkable” properties to a given solution to Fermat equation, the results of Mazur about rational torsion points on elliptic curves, Ribet’s Level Lowering Theorem for modular forms, and the previously mentioned Wiles result.