Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Luis Victor Dieulefait[en] Fermat’s Last Theorem asserts that the equation $x^{n} + y^{n}= x^{n}$ has not non-zero integral solutions $x, y, z$ for $n \geq 3$. In this project we study the Kummer’s proof of the first case of the Theorem along with developing the basic tools of the algebraic number theory. This first case is that the exponent $n$ in the equation is a so-called regular prime and n does not divide any of $x, y, z$.