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In [[number theory]], '''quadratic integers''' are a generalization of the rational [[integer]]s to [[quadratic field]]s. These are [[algebraic integer]]s of the degree [[2 (number)|2]]. Important examples include the [[Gaussian integer]]s and the [[Eisenstein integer]]s. Though they have been studied for more than a hundred years, many open problems remain.

== Definition ==
Quadratic integers are solutions of equations of the form:
:''x''2 + ''Bx'' + ''C'' = 0
for integers ''B'' and ''C''. Such solutions have the form {{math|{{mvar|a}} + ω {{mvar|b}} }}, where {{mvar|a}}, {{mvar|b}} are integers, and where ω is defined by:
:<math>\omega =
\begin{cases}
\sqrt{D} & \mbox{if }D \equiv 2, 3 \pmod{4} \\
{{1 + \sqrt{D}} \over 2} & \mbox{if }D \equiv 1 \pmod{4}
\end{cases}
</math>
({{mvar|D}} is a [[square-free integer]]. Note that the case <math> D \equiv 0\pmod{4} </math> is impossible, since it would imply that D is divisible by 4, a perfect square, which contradicts the fact that D is square-free.).

This characterization was first given by [[Richard Dedekind]] in 1871.<ref>{{harvnb|Dedekind|1871}}, Supplement X, p. 447</ref><ref>{{harvnb|Bourbaki|1994}}, p. 99</ref>

The set of ''all'' quadratic integers is not closed even under [[addition]]. But for any fixed {{mvar|D}} the set of corresponding quadratic integers forms a [[ring (algebra)|ring]], and it is these quadratic integer ''rings'' which are usually studied. Medieval [[Indian mathematics|Indian mathematicians]] had already discovered a [[multiplication]] of quadratic integers of the same {{mvar|D}}, which allows one to solve some cases of [[Pell's equation]]. The study of quadratic integers admits an algebraic version: the study of [[quadratic form#Integral quadratic forms|quadratic forms with integer coefficients]].

== Quadratic integer rings ==

Fixing a square-free integer {{mvar|D}}, the '''quadratic integer ring''' {{math|1='''Z'''[ω] = { {{mvar|a}} + ω {{mvar|b}} : {{mvar|a}}, {{mvar|b}} ∈ '''Z'''} }} is a subring of the [[quadratic field]] <math>\mathbf{Q}(\sqrt{D})</math>. Moreover, '''Z'''[ω] is the [[integral closure]] of '''Z''' in <math>\mathbf{Q}(\sqrt{D})</math>. In other words, it is the [[ring of integers]] <math>\mathcal{O}_{\mathbf{Q}(\sqrt{D})}</math> of '''Q'''({{sqrt|{{mvar|D}}}}) and thus a [[Dedekind domain]]. The quadratic integer rings usually form the first class of examples on which one can build theories, inaccessible in the general case, for example the [[Kronecker–Weber theorem]] in class field theory, see [[#Class number|below]].

=== Examples of complex quadratic integer rings ===
[[Image:Punktraster.svg|thumb|right|105px|Gaussian integers]]
[[Image:EisensteinPrimes-01.svg|thumb|right|120px|Eisenstein primes]]
For {{mvar|D}} < 0, ω is a complex ([[imaginary number|imaginary]] or otherwise non-real) number. Therefore, it is natural to treat a quadratic integer ring as a set of algebraic [[complex number]]s.
* A classic example is <math>\mathbf{Z}[\sqrt{-1}]</math>, the [[Gaussian integer]]s, which was introduced by [[Carl Gauss]] around 1800 to state his biquadratic reciprocity law.<ref>Dummit, pg. 229</ref>
* The elements in <math>\mathcal{O}_{\mathbf{Q}(\sqrt{-3})} = \mathbf{Z}\left[{{1 + \sqrt{-3}} \over 2}\right]</math> are called [[Eisenstein integer]]s.
Both rings mentioned above are rings of integers of [[cyclotomic field]]s '''Q'''(ζ4) and '''Q'''(ζ3) correspondingly.
In contrast, '''Z'''[{{sqrt|−3}}] is not even a [[Dedekind domain]].

=== Examples of real quadratic integer rings ===
For {{mvar|D}} > 0, ω is a [[positive number|positive]] [[irrational number|irrational]] and the corresponding quadratic integer ring is a set of algebraic [[real number]]s. [[Pell's equation]] {{math|1={{mvar|X}}2 − {{mvar|D}} {{mvar|Y}}2 = 1}}, a case of [[Diophantine equation]]s, naturally leads to these rings for {{math|{{mvar|D}} &equiv; 2, 3 (mod 4) }}. Algebraic study of real quadratic integer rings involves determining of the [[unit (ring theory)|invertible elements]] group.
[[Image:Golden spiral in rectangles.svg|thumb|right|Powers of the golden ratio]]
* For {{mvar|D}} = 5, ω is the [[golden ratio]]. A non-negative real number belongs to the ring '''Z'''[(1+{{sqrt|5}})/2] if and only if it can be encoded in [[golden ratio base]] with finite number of 1's. This ring was studied by [[Peter Gustav Lejeune Dirichlet]]. Its invertible elements have the form {{math|±ω{{mvar|n}}}}, where {{mvar|n}} is an arbitrary integer. This ring also arises from studying 5-fold [[rotational symmetry]] on Euclidean plane, for example, [[Penrose tiling]]s.<ref>{{Citation|first=N. G.|last=de Bruijn|journal=Indagationes mathematicae|volume=43|pages=39–66|year=1981|title=Algebraic theory of Penrose's non-periodic tilings of the plane, I, II|url=http://alexandria.tue.nl/repository/freearticles/597566.pdf|format=PDF|issue=1}}</ref>
* Indian mathematician [[Brahmagupta]] treated the equation {{math|1={{mvar|X}}2 − [[61 (number)|61]] {{mvar|Y}}2 = 1}}, where the corresponding ring is '''Z'''[{{sqrt|61}}]. Some results were presented to European community by [[Pierre Fermat]] in 1657.

=== Class number ===

Equipped with the [[field norm|norm]]
:<math>N(a + b\sqrt{D}) = a^2 - Db^2</math>,
<math>\mathcal{O}_{\mathbf{Q}(\sqrt{D})}</math> is an [[Euclidean domain]] (and thus a [[unique factorization domain]], UFD) when {{math|1={{mvar|D}} = −1, −2, −3, −7, −11}}.<ref>Dummit, pg. 272</ref> On the other hand, it turned out that '''Z'''[{{sqrt|−5}}] is not a UFD because, for example, 6 has two distinct factorizations into irreducibles:
:<math>6 = 2(3) = (1 + \sqrt{-5}) (1 - \sqrt{-5}).</math>
(In fact, '''Z'''[{{sqrt|−5}}] has [[Class number (number theory)|class number]] 2.<ref name="class_num">Milne, pg. 64</ref>) The failure of the unique factorization led [[Ernst Kummer]] and Dedekind to develop a theory that would enlarge the set of “prime numbers”; the result was the notion of [[ideal (ring theory)|ideals]] and the decomposition of ideals by [[prime ideal]]s (cf. [[splitting of prime ideals in Galois extensions]]).

Being a Dedekind domain, a quadratic integer ring is a UFD if and only if it is a [[principal ideal domain]] (i.e., its class number is one). However, there are quadratic integer rings that are principal ideal domains but not Euclidean domains. For example, '''Q'''[{{sqrt|−19}}] has class number 1 but its ring of integers is not Euclidean.<ref name="class_num" /> There are effective methods to compute [[ideal class group]]s of quadratic integer rings, but many theoretical questions about their structure are still open after a hundred years.{{as of?|date=February 2012}}

== See also ==
*[[Fermat's Last Theorem]]
*[[Gaussian integer]]
*[[Eisenstein integer]]

== Notes ==
{{reflist}}

== References ==
*{{Bourbaki EHM}}
*{{Citation
| last=Dedekind
| first=Richard
| author-link=Richard Dedekind
| title=Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet
| url=http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN30976923X&DMDID=dmdlog1
| edition=2
| year=1871
| publisher=Vieweg
}}. Retrieved 5. August 2009
*Dummit, D. S., and Foote, R. M., 2004. ''Abstract Algebra'', 3rd ed.
*J.S. Milne. ''[http://www.jmilne.org/math/CourseNotes/ANT301.pdf Algebraic Number Theory]'', Version 3.01, September 28, 2008. online lecture note

[[Category:Algebraic number theory]]
[[Category:Ring theory]]

Locally testable code - Revision history

en>Shorty66 at 13:41, 14 January 2014