Binary quadratic form: Difference between revisions

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=== Examples ===
 
[[Diophantus]] considered whether, for an odd integer <math>n</math>, it is possible to find integers <math>x</math> and <math>y</math> for which <math>n = x^2 + y^2</math>.<ref>{{harvnb|Weil|2001|p.=30}}</ref> When <math>n=65</math>, we have
: <math>\begin{align} 65 &= 1^2 + 8^2,\\
65 &= 4^2 + 7^2,
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"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.
 
"Composition" can also refer to a binary operation on representations of integers by forms. This operation is substantially more complicated{{cn}}<!--<ref>{{harvnb|Shanks|1989}}</ref> full citation not in article yet --> than composition of forms, but arose first historically. We will consider such operations in a separate section below.
 
=== Composing forms and classes ===
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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. But the impact was not immediate. Section V of ''Disquisitiones'' contains truly revolutionary ideas and involves very complicated computations, sometimes left to the reader. Combined, the novelty and complexity made Section V notoriously difficult. [[Dirichlet]] published simplifications of the theory that made it accessible to a broader audience. The culmination of this work is his text ''[[List of important publications in mathematics#Vorlesungen über Zahlentheorie|Vorlesungen über Zahlentheorie]]''. The third edition of this work includes two supplements by [[Dedekind]]. Supplement XI introduces [[ring theory]], and from then on, especially after the 1897 publication of [[Hilbert|Hilbert's]] ''[[List of important publications in mathematics#Zahlbericht|Zahlbericht]]'', the theory of binary quadratic forms lost its preeminent position in [[algebraic number theory]] and became overshadowed by the more general theory of [[algebraic number fields]].
 
Even so, work on binary quadratic forms with integer coefficients continues to the present. This includes numerous results about quadratic number fields, which can often be translated into the language of binary quadratic forms, but also includes developments about forms themselves or that originated by thinking about forms, including [[Daniel Shanks|ShankShanks's]] infrastructure, [[Don Zagier|Zagier's]] reduction algorithm, [[John Horton Conway|Conway's]] topographs, and [[Manjul Bhargava|Bhargava's]] reinterpretation of composition through [[Bhargava cube]]s.
 
==See also==
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==Notes==
{{reflist|30em}}
 
==References==
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| volume=27
}}
* {{Cite bookCitation
| last1=Hardy
| first1=G. H.
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| first2=E. M.
| author2-link=E. M. Wright
| edition={{{edition|6th}}}
| others={{#if: {{{edition|}}}||Revised by [[Roger Heath-Brown|D. R. Heath-Brown]] and [[Joseph H. Silverman|J. H. Silverman]]. Foreword by [[Andrew Wiles]].}}
| title=An Introduction to the Theory of Numbers
| publisher={{#if: {{{edition|}}} | Clarendon Press | [[Oxford University Press]] }}
| ___location=Oxford
| series=
| isbn={{#switch: {{{edition|}}} | 4th = 0-19-853310-1 | 5th = 0-19-853171-0 | 978-0-19-921986-5 }}
| mr={{#switch: {{{edition|}}} | 5th = 0568909 | 2445243 }}
| zbl= {{#switch: {{{edition|}}} | 5th = 0423.10001 | 4th = 0086.25803 | 1159.11001 }}
| year=2008
| year={{#switch: {{{edition|}}} | 4th = 1960 | 5th = 1979 | 2008 }}
| origyear=1938
}}
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==External links==
* [http://oeis.org/wiki/User:Peter_Luschny/BinaryQuadraticForms Peter Luschny, Positive numbers represented by a binary quadratic form]
* {{eom|id=b/b016370|author=A. V. Malyshev|title=Binary quadratic form}}
 
[[Category:Quadratic forms]]