Fermat’s last theorem: work of Kummer, Furtwängler and Terjanian

Abstract

[en] As it is well-known, Fermat’s Last Theorem states that the equation $x^{n} + y^{n} = z^{n}, yxz\neq 0$ 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. This thesis highlights the first steps taken in proving the theorem, before the use of elliptic curves and modularity. Our objective is to resume all these results and try to give a general point of view of what was known before the use of modern methods. Starting with elementary results, we move on to see Kummer’s proof for regular primes. Afterwards, we see how Furtwängler uses class field theory to work on Fermat’s problem, and give us more partial results of the theorem. Finally we study a generalization of Fermat’s last theorem for even exponent, due to Hellegouarch, using again the techniques of class field theory.