It seems that a lot of great mathematicians spent quite a while of their time studying quadratic forms over $\mathbb{Z},\mathbb{Q},\mathbb{Q_p}$ etc. and there is indeed a vast and detailed theory of these. It usually qualifies as part of Number Theory as in Serre's book "A Course In Arithmetic" half of which is devoted to the topic, but it is also claimed to have applications in other areas such as differential topology, finite groups, modular forms (as stated in the preface of the aforementioned book). I would like to have examples of such applications just for general education in order to truly appreciate its universal importance.
Applications of quadratic forms
6
$\begingroup$
big-list
examples-counterexamples
quadratic-forms