Quadratic Forms

An important question in Number Theory is:  which primes can be written as \(x^2 + n y^2\)? This question can be thought of in many different ways: in terms of factorization of primes in algebraic extensions, in terms of norms of number fields but the original viewpoint of Gauss in Disquisionaire Mathematics is in terms of Quadratic Forms.

In the language of Quadratic Forms, the question becomes, which primes can be represented by the quadratic form \(q(x,y) = x^2 + ny^2\). 

The theory of Quadratic Forms revolves around two fundamental problems.

• Representation: Which "numbers" can be represented by a given quadratic form q? 
• Classification: When are two forms "equivalent"? (i.e. can be obtained from each other via an invertible change of variable).