Classification of quadratic forms over Q

The objective of this thesis is the complete classification of quadratic forms over the field of rational numbers. This is achieved by proving the Theorem of Hasse-Minkowski that, given a homogeneous second degree polynomial f, builds a correspondence between the roots of f over the p-adic fields and the roots of f over the rational numbers. The p-adic fields are presented as completions of the rational field with respect to the p-adic absolute values. Using the discriminant and Hasse-Minkowski invariant, a complete classification of the quadratic forms over the p-adic fields is achieved. From these results it is possible to find whether or not two quadratic forms are equivalent over the rational field by computing countable invariants.