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Pointed out a mistake; but, unfortunately, don't know what would be good to put in its place. |
thanks. fix problem (one direction is true), and add ref to Cohen's book |
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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''' consists of finitely many classes of forms.
An integral binary quadratic form is called '''primitive''' if ''a'', ''b'', and ''c'' have no common factor.
From a modern perspective, the class group of a fundamental discriminant ''D'' is [[isomorphic]] to the [[narrow class group]] of the [[quadratic field]] <math>\mathbf{Q}(\sqrt{D})</math> of discriminant ''D''.<ref>{{harvnb|Fröhlich|Taylor|1993|loc=Theorem 58}}</ref> For negative ''D'', the narrow class group is the same as the [[ideal class group]], but for positive ''D'' it may twice as big.
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* Johannes Buchmann, Ulrich Vollmer: ''Binary Quadratic Forms'', Springer, Berlin 2007, ISBN 3-540-46367-4
* Duncan A. Buell: ''Binary Quadratic Forms'', Springer, New York 1989
* {{Citation
| last=Cohen
| first=Henri
| author-link=Henri Cohen (number theorist)
| title=A Course in Computational Algebraic Number Theory
| publisher=[[Springer-Verlag]]
| ___location=Berlin, New York
| series=Graduate Texts in Mathematics
| isbn=978-3-540-55640-4
| id={{MR|1228206}}
| year=1993
| volume=138
}}
* {{Citation
| last= Fröhlich
| |||