Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2017, Director: Xavier Guitart Morales, 2017
We construct the field of p-adic numbers as the completion of the rational numbers with respect to the p-adic distance. We also give the definition of the p-adic $\zeta$ function: a function of a p-adic variable that interpolates the values of Riemann’s $\zeta$ function at negative integers, and which we construct using the theory of p-adic integration.