# Kleinian integer

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In [mathematical cryptography](/source/mathematical_cryptography), a '''Kleinian integer''' is a [complex number](/source/complex_number) of the form <math>m+n\frac{1+\sqrt{-7}}{2}</math>, with ''m'' and ''n'' [rational integers](/source/Integer). They are named after [Felix Klein](/source/Felix_Klein).

The Kleinian integers form a [ring](/source/Ring_(mathematics)) called the '''Kleinian ring''', which is the [ring of integers](/source/ring_of_integers) in the imaginary [quadratic field](/source/quadratic_field) <math>\mathbb{Q}(\sqrt{-7})</math>. This ring is a [unique factorization domain](/source/unique_factorization_domain).

==See also==

*[Eisenstein integer](/source/Eisenstein_integer)
*[Gaussian integer](/source/Gaussian_integer)

==References==

* {{Citation|authorlink=John Horton Conway|last1=Conway|first1=John Horton|last2=Smith|first2=Derek A.|year=2003|title=On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry|publisher=A. K. Peters, Ltd.|isbn=978-1-56881-134-5|postscript=<!--none-->}}. ([http://nugae.wordpress.com/2007/04/25/on-quaternions-and-octonions/ Review]).
*{{citation|series=Lecture Notes in Computer Science |volume= 4249|year= 2006|pages=445–459
|title=Cryptographic Hardware and Embedded Systems - CHES 2006|chapter=FPGA Implementation of Point Multiplication on Koblitz Curves Using Kleinian Integers
|first1=V. S.|last1= Dimitrov|first2= K. U.|last2= Järvinen|first3= M. J.|last3= Jacobson|first4= W. F.|last4= Chan|first5= Z.|last5= Huang|doi=10.1007/11894063_35|isbn=978-3-540-46559-1|doi-access= free}}

<!--Category:Algebraic numbers-->Category:Quadratic irrational numbers
Category:Ring theory

{{numtheory-stub}}

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