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Two integral forms are called '''equivalent''' if there exists an invertible integral linear change of variables that transforms the first form into the second. This defines an [[equivalence relation]] on the set of integral quadratic forms, whose elements are called '''classes''' of quadratic forms. Equivalent forms necessarily have the same [[discriminant of a quadratic form|discriminant]]
: <math> D(f)=b^2-4ac, \quad D(f)\equiv 0,1
Gauss proved that for every value ''D'', there are only finitely many classes of binary quadratic forms with discriminant ''D''. Their number is the '''class number''' of discriminant ''D''. He described an algorithm, called '''reduction''', for constructing a canonical representative in each class, the '''reduced form''', whose coefficients are the smallest in a suitable sense. One of the deepest discoveries of Gauss was the existence of a natural '''composition law''' on the set of classes of binary quadratic forms of given discriminant, which makes this set into a finite [[abelian group]] called the '''class group''' of discriminant ''D''. Gauss also considered a coarser notion of equivalence, under which the set of binary quadratic forms of a fixed discriminant splits into several genera of forms and each '''[[Genus of a quadratic form|genus]]''' consists of finitely many classes of forms.
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