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In [[number theory]], '''quadratic integers''' are a generalization of the [[integer]]s to [[quadratic field]]s.  Quadratic integers are [[algebraic integer]]s of degree two, that is, solutions of equations of the form 
:{{math|1=''x''<sup>2</sup> + ''Bx'' + ''C'' = 0}}
with {{math|''B''}} and {{math|''C''}} integers. When algebraic integers are considered, the usual integers are often called ''rational integers''.

Common examples of quadratic integers are the square roots of integers, such as {{math|{{sqrt|2}}}}, and the [[complex number]] {{math|1=''i'' = {{sqrt|–1}}}}, which generates the [[Gaussian integer]]s. Another common example is the non-real cubic [[root of unity]] {{math|{{sfrac|−1 + {{sqrt|–3}}|2}}}}, which generates the [[Eisenstein integer]]s.

Quadratic integers occur in the solutions of many [[Diophantine equation]]s, such as [[Pell's equation]]s, and other questions related to integral [[quadratic forms]]. The study of '''rings of quadratic integers''' is basic for many questions of [[algebraic number theory]].

==History==
{{expand section|date=March 2015}}

Medieval [[Indian mathematics|Indian mathematicians]] had already discovered a multiplication of quadratic integers of the same {{mvar|D}}, which allowed them to solve some cases of [[Pell's equation]].{{citation needed|date=March 2015}}

The characterization{{clarify|reason=which characterization?|date=March 2015}} of the quadratic integers was first given by [[Richard Dedekind]] in 1871.<ref>{{harvnb|Dedekind|1871}}, Supplement X, p. 447</ref><ref>{{harvnb|Bourbaki|1994}}, p. 99</ref>

== Definition ==
A '''quadratic integer''' is an [[algebraic integer]] of degree two. More explicitly, it is a [[complex number]] <math> x=\frac{-B\pm\sqrt{B^2-4C}}{2}</math>, which solves an equation of the form {{math|1=''x''<sup>2</sup> + ''Bx'' + ''C'' = 0}}, with ''B'' and ''C'' [[integer]]s. Each quadratic integer that is not an integer is not rational—namely, it's a real [[irrational number]] if {{math|''B''<sup>2</sup> – 4''C'' > 0}} and non-real if {{math|''B''<sup>2</sup> – 4''C'' < 0}}—and lies in a uniquely determined [[quadratic field]] <math>\mathbb{Q}(\sqrt{D})</math>, the extension of <math>\mathbb{Q}</math> generated by the square-root of the unique [[square-free integer]] {{math|''D''}} that satisfies {{math|1=''B''<sup>2</sup> – 4''C'' = ''DE''<sup>2</sup>}} for some integer {{math|''E''}}. {{math|''D''}} may be positive or negative.

The quadratic integers (including the ordinary integers), which belong to a quadratic field <math>\mathbb{Q}(\sqrt{D})</math>, form an [[integral domain]] called the ''ring of integers of'' <math>\mathbb{Q}(\sqrt{D}).</math>

Here and in the following, {{math|''D''}} is supposed to be a [[square-free integer]]. This does not restrict the generality, as the equality {{math|1={{sqrt|''a''<sup>2</sup>''D''}} = ''a''{{sqrt|''D''}}}} (for any positive integer {{math|''a''}}) implies <math>\mathbb{Q}(\sqrt{D})=\mathbb{Q}(\sqrt{a^2D}).</math>

Every quadratic integer may be written {{math|''a'' + ''ωb'' }}, where {{math|''a''}} and {{math|''b''}} are integers, and where {{math|''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>
(as {{math|''D''}} has been supposed square-free the case <math> D \equiv 0\pmod{4} </math> is impossible, since it would imply that D would be divisible by the square 4).<ref>{{Cite web|url=https://math.stackexchange.com/q/1198188 |title=Why is quadratic integer ring defined in that way?|website=math.stackexchange.com|access-date=2016-12-31}}</ref>

Although the quadratic integers belonging to a given quadratic field form a [[ring (mathematics)|ring]], the set of ''all'' quadratic integers is not a ring, because it is not closed under [[addition]], as <math> \sqrt{2}+\sqrt{3}</math> is an algebraic integer, which has a [[minimal polynomial (field theory)|minimal polynomial]] of degree four.

==Norm and conjugation ==
A quadratic integer in <math>\mathbb{Q}(\sqrt{D})</math> may be written 
:{{math|''a'' + ''b''{{sqrt|''D''}}}},
where {{math|''a''}} and {{math|''b''}} are either both integers, or, only if {{math|''D'' ≡ 1 (mod 4)}}, both [[half-integer|halves of odd integers]]. The '''norm''' of such a quadratic integer is 
:{{math|1=''N''(''a'' + ''b''{{sqrt|''D''}}) = ''a''<sup>2</sup> – ''Db''<sup>2</sup>}}.

The norm of a quadratic integer is always an integer. If {{math|''D'' < 0}}, the norm of a quadratic integer is the square of its [[absolute value]] as a complex number (this is false if {{math|''D'' > 0}}). The norm is a [[completely multiplicative function]], which means that the norm of a product of quadratic integers is always the product of their norms.

Every quadratic integer {{math|''a'' + ''b''{{sqrt|''D''}}}} has a '''conjugate'''
:<math>\overline{a+b\sqrt{D}} = a-b\sqrt{D}.</math>
A quadratic integer has the same norm as its conjugate, and this norm is the product of the quadratic integer and its conjugate. The conjugate of a sum or a product of quadratic integers is the sum or the product (respectively) of the conjugates. This means that the conjugation is an [[automorphism]] of the ring of the integers of <math>\mathbb{Q}(\sqrt{D})</math>—see {{slink||Quadratic integer rings}}, below.

== Quadratic integer rings ==

Every [[square-free integer]] (different from 0 and 1) {{math|''D''}} defines a '''quadratic integer ring''', which is the [[integral domain]] consisting of the [[algebraic integer]]s contained in <math>\mathbf{Q}(\sqrt{D}).</math> It is the set {{math|1=  '''Z'''[''ω''] = {''a'' + ''ωb'' : ''a'', ''b'' ∈ '''Z'''},}} where <math>\omega = \tfrac{1+\sqrt D}{2}</math> if {{math|1=''D'' = 4''k'' +1}}, and {{math|1=''ω'' = {{sqrt|''D''}}}} otherwise. It is often denoted <math>\mathcal{O}_{\mathbf{Q}(\sqrt{D})}</math>, because it is  the [[ring of integers]] of {{math|'''Q'''({{sqrt|''D''}}}}), which is the [[integral closure]] of {{math|'''Z'''}} in <math>\mathbf{Q}(\sqrt{D}).</math> The ring {{math|'''Z'''[''ω'']}} consists of all roots of all equations {{math|1=''x''<sup>2</sup> + ''Bx'' + ''C'' = 0}} whose [[discriminant]] {{math|1=''B''<sup>2</sup> − 4''C''}} is the product of {{mvar|D}} by the square of an integer. In particular {{sqrt|''D''}} belongs to {{math|'''Z'''[''ω'']}}, being a root of the  equation {{math|1=''x''<sup>2</sup> − ''D'' = 0}}, which has {{math|4''D''}} as its discriminant. 

The [[square root]] of any integer is a quadratic integer, as every integer can be written {{math|1=''n'' = ''m''<sup>2</sup>''D''}}, where {{mvar|D}} is a square-free integer, and its square root is a root of {{math|1=''x''<sup>2</sup> − ''m''<sup>2</sup>''D'' = 0}}.

The [[fundamental theorem of arithmetic]] is not true in many rings of quadratic integers. However, there is a unique factorization for [[ideal (ring theory)|ideals]], which is expressed by the fact that every ring of algebraic integers is a [[Dedekind domain]]. Being the simplest examples of algebraic integers, quadratic integer are commonly the starting examples of most studies of [[algebraic number theory]].<ref>M. Artin, ''Algebra'' (2nd ed) Ch 13</ref>

The quadratic integer rings divide in two classes depending on the sign of {{math|''D''}}. If {{math|''D'' > 0}}, all elements of <math>\mathcal{O}_{\mathbf{Q}(\sqrt{D})}</math> are real, and the ring is a ''real quadratic integer ring''. If {{math|''D'' < 0}}, the only real elements of <math>\mathcal{O}_{\mathbf{Q}(\sqrt{D})}</math> are the ordinary integers, and the ring is a ''complex quadratic integer ring''. 

For real quadratic integer rings, the [[ideal class number]], which measures the failure of unique factorization, is given in [https://oeis.org/A003649 OEIS A003649]; for the imaginary case, they are given in [https://oeis.org/A000924 OEIS A000924].

===Units===
A quadratic integer is a [[unit (ring theory)|unit]] in the ring of the integers of <math>\mathbb{Q}(\sqrt{D})</math> if and only if its norm is {{math|1}} or {{math|–1}}. In the first case its [[multiplicative inverse]] is its conjugate. It is the negation of its conjugate in the second case.

If {{math|''D'' < 0}}, the ring of the integers of <math>\mathbb{Q}(\sqrt{D})</math> has at most six units. In the case of the [[Gaussian integer]]s ({{math|1=''D'' = –1}}), the four units are {{math|1, –1, {{sqrt|–1}}, –{{sqrt|–1}}}}. In the case of the [[Eisenstein integer]]s ({{math|1=''D'' = –3}}), the six units are {{math|±1, {{sfrac|±1 ± {{sqrt|–3}}|2}}}}. For all other negative {{math|''D''}}, there are only two units, which are {{math|1}} and {{math|–1}}.

If {{math|''D'' > 0}}, the ring of the integers of <math>\mathbb{Q}(\sqrt{D})</math> has infinitely many units that are equal to {{math|±''u''<sup>''i''</sup>}}, where {{math|''i''}} is an arbitrary integer, and {{math|''u''}} is a particular unit called a ''[[fundamental unit (number theory)|fundamental unit]]''. Given a fundamental unit {{math|''u''}}, there are three other fundamental units, its conjugate <math>\overline{u},</math> and also <math>-u</math> and <math>-\overline{u}.</math> Commonly, one calls ''the'' fundamental unit, the unique one which has an absolute value greater than 1 (as a real number). It is the unique fundamental unit that may be written as {{math|''a'' + ''b''{{sqrt|''D''}}}}, with {{math|''a''}} and {{math|''b''}} positive (integers or halves of integers).

The fundamental units for the 10 smallest positive square-free {{math|''D''}} are {{math|1 + {{sqrt|2}}}}, {{math|2 + {{sqrt|3}}}}, {{math|{{sfrac|1 + {{sqrt|5}}|2}}}} (the [[golden ratio]]), {{math|5 + 2{{sqrt|6}}}}, {{math|8 + 3{{sqrt|7}}}}, {{math|3 + {{sqrt|10}}}}, {{math|10 + 3{{sqrt|11}}}}, {{math|{{sfrac|3 + {{sqrt|13}}|2}}}}, {{math|15 + 4{{sqrt|14}}}}, {{math|4 + {{sqrt|15}}}}. For larger {{math|''D''}}, the coefficients of the fundamental unit may be very large. For example, for {{math|1=''D'' = 19, 31, 43}}, the fundamental units are respectively 
{{math|170 + 39 {{sqrt|19}}}}, {{math|1520 + 273 {{sqrt|31}}}} and {{math|3482 + 531 {{sqrt|43}}}}.

=== 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}}&nbsp;&lt;&nbsp;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'''(ζ<sub>4</sub>) and '''Q'''(ζ<sub>3</sub>) correspondingly.
In contrast, '''Z'''[{{sqrt|−3}}] is not even a [[Dedekind domain]].

Both above examples are [[principal ideal ring]]s and also [[Euclidean domain]]s for the norm. This is not the case for 
:<math>\mathcal{O}_{\mathbf{Q}(\sqrt{-5})} = \mathbf{Z}\left[\sqrt{-5}\right],</math>
which is not even a [[unique factorization domain]]. This can be shown as follows.

In <math>\mathcal{O}_{\mathbf{Q}(\sqrt{-5})},</math> we have
:<math>9 = 3\cdot3 = (2+\sqrt{-5})(2-\sqrt{-5}).</math>
The factors 3, <math>2+\sqrt{-5}</math> and <math>2-\sqrt{-5}</math> are [[irreducible element|irreducible]], as they have all a norm of 9, and if they were not irreducible, they would have a factor of norm 3, which is impossible, the norm of an element different of {{math|±1}} being at least 4. Thus the factorization of 9 into irreducible factors is not unique.

The [[ideal (ring theory)|ideals]] <math>\langle 3, 1+\sqrt{-5}\rangle</math> and <math>\langle 3, 1-\sqrt{-5}\rangle</math> are not [[principal ideal|principal]], as a simple computation shows that their product is the ideal generated by 3, and, if they were principal, this would imply that 3 would not be irreducible.

=== Examples of real quadratic integer rings ===
[[Image:Golden spiral in rectangles.svg|thumb|right|Powers of the golden ratio]]
For {{math|''D'' > 0}}, {{math|ω}} is a positive [[irrational number|irrational]] real number, and the corresponding quadratic integer ring is a set of algebraic [[real number]]s. The solutions of the [[Pell's equation]] {{math|1=''X''<sup>2</sup> − ''D'' ''Y''<sup>2</sup> = 1}}, a [[Diophantine equation]] that has been widely studied, are the [[unit (ring theory)|units]] of these rings, for {{math|''D'' ≡ 2, 3 (mod 4)}}. 
* For {{math|1=''D'' = 5}}, {{math|1=ω = {{sfrac|1+{{sqrt|5}}|2}}}} is the [[golden ratio]]. This ring was studied by [[Peter Gustav Lejeune Dirichlet]]. Its units have the form {{math|±ω<sup>''n''</sup>}}, where {{math|''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|issue=1}}</ref>
* Indian mathematician [[Brahmagupta]] treated the Pell's equation {{math|1=''X''<sup>2</sup> − 61 ''Y''<sup>2</sup> = 1}},  corresponding  to the ring is {{math|'''Z'''[{{sqrt|61}}]}}. Some results were presented to European community by [[Pierre Fermat]] in 1657.{{which|date=May 2015}}

=== Principal rings of quadratic integers ===

[[Unique factorization]] property  is not always verified for rings of quadratic integers, as seen above for the case of {{math|'''Z'''[{{sqrt|−5}}]}}. However, as for every [[Dedekind domain]], a ring of quadratic integers is a [[unique factorization domain]] if and only if it is a [[principal ideal domain]]. This occurs if and only if the [[ideal class group|class number]] of the corresponding [[quadratic field]]  is one.

The imaginary rings of quadratic integers that are principal ideal rings have been completely determined. These are <math>\mathcal{O}_{\mathbf{Q}(\sqrt{D})}</math> for 
:{{math|1=''D'' = −1, −2, −3, −7, −11, −19, −43, −67, −163}}. 
This result was first conjectured by [[Gauss]] and proven by [[Kurt Heegner]], although Heegner's proof was not believed until [[Harold Stark]] gave a later proof in 1967. (See [[Stark–Heegner theorem]].) This is a special case of the famous [[class number problem]].

There are many known positive integers {{math|''D'' > 0}}, for which the ring of quadratic integers is a principal ideal ring. However, the complete list is not known; it is not even known if the number of these principal ideal rings is finite or not.

=== Euclidean rings of quadratic integers ===
{{see also|Euclidean domain#Norm-Euclidean fields}}
When a ring of quadratic integers is a [[principal ideal domain]], it is interesting to know whether it is a [[Euclidean domain]]. This problem has been completely solved as follows.

Equipped with the norm
<math>N(a + b\sqrt{D}) = a^2 - Db^2</math> as a [[Euclidean function]],
<math>\mathcal{O}_{\mathbf{Q}(\sqrt{D})}</math> is a Euclidean domain for negative {{math|''D''}} when 
:{{math|1=''D'' = −1, −2, −3, −7, −11}},<ref>Dummit, pg. 272</ref> 
and, for positive {{math|''D''}}, when 
:{{math|1=''D'' = 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73}} {{OEIS|id=A048981}}.
There is no other ring of quadratic integers that is Euclidean with the norm as a Euclidean function.<ref>{{cite book | last = LeVeque | first = William J. | authorlink = William J. LeVeque | title = Topics in Number Theory, Volumes I and II | publisher = Dover Publications | location = New York | year = 2002 | origyear = 1956 | isbn = 978-0-486-42539-9 | zbl=1009.11001 | pages=II:57,81}}</ref>

For negative {{math|''D''}}, a ring of quadratic integers is Euclidean if and only if the norm is a [[Euclidean function]] for it. It follows that, for 
:{{math|1=''D'' = −19, −43, −67, −163}}, 
the four corresponding rings of quadratic integers are among the rare known examples of principal ideal domains that are not Euclidean domains.

On the other hand, the [[generalized Riemann hypothesis]] implies that a ring of ''real'' quadratic integers that is a principal ideal domain is also a Euclidean domain for some Euclidean function, which can indeed differ from the usual norm.<ref>P. Weinberger, ''On Euclidean rings of algebraic integers''. In: Analytic Number Theory (St. Louis, 1972), Proc. Sympos. Pure Math. 24(1973), 321–332.</ref>
The values ''D'' = 14, 69 were the first for which the ring of quadratic integers was proven to be Euclidean, but not norm-Euclidean.<ref>M. Harper, <math>\mathbb {Z}[\sqrt{14}]</math> ''is Euclidean''. Can. J. Math. 56(2004), 55–70.</ref><ref>David A. Clark, ''A quadratic field which is Euclidean but not norm-Euclidean'', Manuscripta Mathematica, '''83'''(1994), 327–330 [http://www.math.clemson.edu/~jimlb/CourseNotes/AbstractAlgebra/EuclideanNotNormEuclidean.pdf]</ref>
<!-- material misplaced here, but maybe useful for the preceding section (I will see that later : 
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 introduction of the notion of ideal, and the definition of what is now called a [[Dedekind domain]]: All the ring of integers of number fields are Dedekind domains, and the ideals of a Dedekind domain have the property of unique factorization into products of [[prime ideal]]s.

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}}
-->

== 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.
*Artin, M, ''Algebra'', 2nd ed., Ch 13.

==Further reading==

*J.S. Milne. ''[http://www.jmilne.org/math/CourseNotes/ANT301.pdf Algebraic Number Theory]'', Version 3.01, September 28, 2008. online lecture notes

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