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Introduction and definition of composition |
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where ''n'' is an integer. If we consider the class of <math>Ax^2 + Bxy + Cy^2</math> under this action, the middle coefficients of the forms in the class form a congruence class of integers modulo 2''A''. Thus, composition gives a well-defined function from pairs of binary quadratic forms to such classes.
It can be shown that if <math>f_1</math> and <math>f_2</math> are equivalent to <math>g_1</math> and <math>g_2</math> respectively, then the composition of <math>f_1</math> and <math>f_2</math> is equivalent to the composition of <math>g_1</math> and <math>g_2</math>. It follows that composition induces a well-defined operation on primitive classes of discriminant <math>\Delta</math>, and as mentioned above, Gauss showed these classes form a finite abelian group. The [[identity element|identity]] class in the group is the unique class containing all forms <math>x^2 + Bxy + Cy^2</math>, i.e., with first coefficient 1. (It can be shown that all such forms lie in a single class, and the restriction <math>\Delta \equiv 0 \text{ or } 1 \pmod{4}</math> implies that there exists such a form of every discriminant.) To [[inverse element|invert]] a class, we take a representative <math>Ax^2 + Bxy + Cy^2</math> and form the class either of <math>Ax^2 - Bxy + Cy^2</math>. Alternatively, we can form the class of <math>Cx^2 + Bxy + Ay^2</math> since this and <math>Ax^2 - Bxy + Cy^2</math> are equivalent.
== Genera of binary quadratic forms ==
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