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| In [[mathematics]], a '''binary quadratic form''' is a [[quadratic form]] in two variables. More concretely, it is a [[homogeneous polynomial]] of degree 2 in two variables
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| : <math> q(x,y)=ax^2+bxy+cy^2, \, </math>
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| where ''a'', ''b'', ''c'' are the '''coefficients'''. Properties of binary quadratic forms depend in an essential way on the nature of the coefficients, which may be [[real number]]s, [[rational number]]s, or in the most delicate case, [[integer]]s. Arithmetical aspects of the theory of binary quadratic forms are related to the arithmetic of [[quadratic field]]s and have been much studied, notably, by [[Gauss]] in Section V of ''[[Disquisitiones Arithmeticae]]''. The theory of binary quadratic forms has been extended in two directions: general [[number field]]s and quadratic forms in ''n'' variables.
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| ==Brief history==
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| Binary quadratic forms were considered already by [[Fermat]], in particular, in the question of [[Fermat's theorem on sums of two squares|representations of numbers as sums of two squares]]. The theory of [[Pell's equation]] may be viewed as a part of the theory of binary quadratic forms. [[Lagrange]] in 1773 initiated the development of the general theory of quadratic forms. First systematic treatment of binary quadratic forms is due to [[Adrien-Marie Legendre|Legendre]]. Their theory was advanced much further by [[Carl Friedrich Gauss|Gauss]] in ''[[Disquisitiones Arithmeticae]]''. He considered questions of equivalence and reduction and introduced composition of binary quadratic forms (Gauss and many subsequent authors wrote 2''b'' in place of ''b''; the modern convention allowing the coefficient of ''xy'' to be odd is due to [[Gotthold Eisenstein|Eisenstein]]). These investigations of Gauss strongly influenced both the arithmetical theory of quadratic forms in more than two variables and the subsequent development of algebraic number theory, where quadratic fields are replaced with more general [[number field]]s.
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| ==Main questions==
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| A classical question in the theory of integral quadratic forms (those with integer coefficients) is the '''representation problem''': describe the set of numbers represented by a given quadratic form ''q''. If the number of representations is finite then a further question is to give a closed formula for this number. The notion of ''equivalence'' of quadratic forms and the related ''reduction theory'' are the principal tools in addressing these questions.
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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]]
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| : <math> D(f)=b^2-4ac, \quad D(f)\equiv 0,1\, (\!\!\!\!\! \mod 4). </math> | |
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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 of a quadratic form|genus]]''' consists of finitely many classes of forms.
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| An integral binary quadratic form is called '''primitive''' if ''a'', ''b'', and ''c'' have no common factor. If a form's discriminant is a [[fundamental discriminant]], then the form is primitive.<ref>{{harvnb|Cohen|1993|loc=§5.2}}</ref>
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| 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 be twice as big.
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| ==See also==
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| * [[Legendre symbol]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * Johannes Buchmann, Ulrich Vollmer: ''Binary Quadratic Forms'', Springer, Berlin 2007, ISBN 3-540-46367-4
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| * Duncan A. Buell: ''Binary Quadratic Forms'', Springer, New York 1989
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| * {{Citation
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| | last=Cohen
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| | first=Henri
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| | author-link=Henri Cohen (number theorist)
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| | title=A Course in Computational Algebraic Number Theory
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| | publisher=[[Springer-Verlag]]
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| | location=Berlin, New York
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| | series=Graduate Texts in Mathematics
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| | isbn=978-3-540-55640-4
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| | mr=1228206
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| | year=1993
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| | volume=138
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| }}
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| * {{Citation
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| | last= Fröhlich
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| | first = Albrecht
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| | authorlink= Albrecht Fröhlich
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| | last2=Taylor
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| | first2=Martin
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| | authorlink2= Martin J. Taylor
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| | title=Algebraic number theory
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| | publisher=[[Cambridge University Press]]
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| | series=Cambridge Studies in Advanced Mathematics
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| | isbn=978-0-521-43834-6
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| | mr=1215934
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| | year=1993
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| | volume=27
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| }}
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| ==External links==
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| * {{eom|id=b/b016370|author=A.V.Malyshev|title=Binary quadratic form}}
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| [[Category:Quadratic forms]]
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