Binary quadratic form: Difference between revisions

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== Composition ==

'''Composition''' most commonly refers to a [[binary operation]] on primitive equivalence classes of forms of the same discriminant., Oneone of the deepest discoveries of Gauss , which makes this set into a finite [[abelian group]] called the '''form class group''' (or simply class group) of discriminant <math>\Delta</math>. [[Class group]]s have since become one of the central ideas in algebraic number theory. From a modern perspective, the class group of a fundamental discriminant <math>\Delta</math> is [[isomorphic]] to the [[narrow class group]] of the [[quadratic field]] <math>\mathbf{Q}(\sqrt{\Delta})</math> of discriminant <math>\Delta</math>.<ref>{{harvnb|Fröhlich|Taylor|1993|loc=Theorem 58}}</ref> For negative <math>\Delta</math>, the narrow class group is the same as the [[ideal class group]], but for positive <math>\Delta</math> it may be twice as big.

"Composition" also sometimes refers to, roughly, a binary operation on binary quadratic forms. The word "roughly" indicates two caveats: only certain pairs of binary quadratic forms can be composed, and the resulting form is not well-defined (although its equivalence class is). The composition operation on equivalence classes is defined by first defining composition of forms and then showing that this induces a well-defined operation on classes.