Introduzione alla teoria dei numeri p-adici e teorema di Hasse-Minkowski

The tool of p-adic numbers is of central importance in modern number theory. Given the field Q of rational numbers, one can equip it with the norm induced by the p-adic valuation modulo a prime number p; the field Qp of p-adic numbers modulo p is the metric completion of Q with regard to that norm. In this work we carefully build the theory of p-adic numbers, both from an algebraic and analytic point of view, emphasizing their role in the arithmetic of Q. We then focus on the quadratic properties of Qp, culminating with the proof of the notorious Hasse-Minkowski theorem, which states that a quadratic form in Q has non-trivial solutions if and only if it has non-trivial solutions over the field of real numbers R and over Qp for every prime number p. This is a great example of what is known as local-global principle: to check wether a property holds globally (in our case over Q) one checks wether it holds locally at every point (in our case over Qp). This approach is of central importance the modern theory of local and global fields.