View source for Quadratic integer

{{short description|Root of a quadratic polynomial with a unit leading coefficient}}
In [[number theory]], '''quadratic integers''' are a generalization of the usual [[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 {{mvar|b}} and {{mvar|c}} (usual) integers. When algebraic integers are considered, the usual integers are often called ''rational integers''.

Common examples of quadratic integers are the [[square root]]s of rational 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 number|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 form]]s. 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 given in ''{{slink||Explicit representation}}'' of the quadratic integers was first given by [[Richard Dedekind]] in 1871.{{sfn|Dedekind|1871|loc=Supplement X, p. 447|ps=}}{{sfn|Bourbaki|1994|p=99|ps=}}

== Definition ==
A '''quadratic integer''' is an [[algebraic integer]] of degree two. More explicitly, it is a [[complex number]] <math>x = (-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 {{math|''b''}} and {{math|''c''}} [[integer]]s. Each quadratic integer that is not an integer is not [[rational number|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]] {{mvar|D}} that satisfies {{math|1=''b''<sup>2</sup> − 4''c'' = ''De''<sup>2</sup>}} for some integer {{math|''e''}}. If {{math|''D''}} is positive, the quadratic integer is real. If {{math|''D'' < 0}}, it is ''imaginary'' (that is, complex and non-real).

The quadratic integers (including the ordinary integers) that 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>

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 or multiplication. For example, <math>1+\sqrt{2}</math> and <math>\sqrt{3}</math> are quadratic integers, but <math> 1+\sqrt{2}+\sqrt{3}</math> and <math>(1+\sqrt{2})\cdot\sqrt{3}</math> are not, as their [[minimal polynomial (field theory)|minimal polynomials]] have [[degree of a polynomial|degree]] four.

== Explicit representation ==
Here and in the following, the quadratic integers that are considered belong to a [[quadratic field]] <math>\mathbb{Q}(\sqrt{D}\,),</math> where {{mvar|D}} is a square-free integer. This does not restrict the generality, as the equality {{math|1={{sqrt|''a''<sup>2</sup>''D''}} = ''a''&hairsp;{{sqrt|''D''}}}} (for any positive integer {{mvar|a}}) implies <math>\mathbb{Q}(\sqrt{D}\,) = \mathbb{Q}(\sqrt{a^2D}\,).</math>

An element {{mvar|x}} of <math>\mathbb{Q}(\sqrt{D}\,)</math> is a quadratic integer if and only if there are two integers {{mvar|a}} and {{mvar|b}} such that either 
: <math>x = a+b\sqrt D,</math> 
or, if {{math|''D''&nbsp;−&thinsp;1}} is a multiple of {{math|4}}
: <math>x = \frac{a}{2}+\frac{b}{2}\sqrt{D},</math> with {{mvar|a}} and {{mvar|b}} both [[parity (mathematics)|odd]]

In other words, every quadratic integer may be written {{math|''a'' + ''ωb'' }}, where {{mvar|a}} and {{mvar|b}} are integers, and where {{mvar|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 {{mvar|D}} has been supposed square-free the case <math display="inline">D \equiv 0\pmod{4}</math> is impossible, since it would imply that {{mvar|D}} is divisible by the [[square number|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>

== Norm and conjugation ==
A quadratic integer in <math>\mathbb{Q}(\sqrt{D}\,)</math> may be written 
: {{math|''a'' + ''b''{{sqrt|''D''}}}},
where {{mvar|a}} and {{mvar|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''&hairsp;(''a'' + ''b''{{sqrt|''D''}}&hairsp;) = ''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) {{mvar|D}} defines a '''quadratic integer ring''', which is the [[integral domain]] consisting of the [[algebraic integer]]s contained in <math>\mathbb{Q}(\sqrt{D}\,).</math> It is the set <math> \mathbb{Z}[\omega] = \{ a + \omega b : a,b \in \mathbb{Z} \} </math> 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}_{\mathbb{Q}(\sqrt{D}\,)}</math>, because it is  the [[ring of integers]] of <math>\mathbb{Q}(\sqrt{D}\,)</math>, which is the [[integral closure]] of <math>\mathbb{Z}</math> in <math>\mathbb{Q}(\sqrt{D}\,).</math> The ring <math> \mathbb{Z}[\omega]</math> consists of all [[root of a polynomial|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>\mathbb{Z}[\omega]</math>, 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 integers are commonly the starting examples of most studies of [[algebraic number theory]].{{sfn|Artin|loc=Ch 13|ps=}}

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

For real quadratic integer rings, the [[class number (number theory)|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 [[additive inverse|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 {{mvar|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|±&hairsp;''u''<sup>''i''</sup>}}, where {{mvar|i}} is an arbitrary integer, and {{mvar|u}} is a particular unit called a ''[[fundamental unit (number theory)|fundamental unit]]''. Given a fundamental unit {{mvar|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 {{mvar|a}} and {{mvar|b}} positive (integers or halves of integers).

The fundamental units for the 10 smallest positive square-free {{mvar|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 {{mvar|D}}, the [[coefficient]]s 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;<&nbsp;0, {{mvar|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.{{sfn|Dummit|Foote|2004|p=229|ps=}}
* 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}}, {{mvar|ω}} is a positive irrational [[real number]], and the corresponding quadratic integer ring is a set of algebraic real numbers. The solutions of the [[Pell's equation]] {{math|1=''X''<sup>&thinsp;2</sup> − ''DY''<sup>&thinsp;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 {{mvar|n}} is an arbitrary integer. This ring also arises from studying 5-fold [[rotational symmetry]] on Euclidean plane, for example, [[Penrose tiling]]s.{{sfn|de Bruijn|1981|ps=}}
* Indian mathematician [[Brahmagupta]] treated the Pell's equation {{math|1={{itco|''X''}}<sup>2</sup> − 61{{itco|''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 ===

The [[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 [[conjecture]]d by [[Gauss]] and [[mathematical proof|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 {{mvar|D}} when 
: {{math|1=''D'' = −1, −2, −3, −7, −11}},{{sfn|Dummit|Foote|2004|p=272|ps=}} 
and, for positive {{mvar|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.{{sfn|LeVeque|2002|pp=II:57,81|ps=}}
For negative {{mvar|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 {{nowrap|1=''D'' = 14, 69}} were the first for which the ring of quadratic integers was proven to be Euclidean, but not norm-Euclidean.{{sfn|Harper|2004|ps=}}{{sfn|Clark|1994|ps=}}
<!-- 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 ==
{{refbegin}}
* {{citation
  | last1=Artin | first1=M
  | title=Algebra
  | edition=2nd
  }}
* {{Bourbaki EHM}}
* {{citation
  | last1=Clark | first1=David A.
  | year=1994
  | title=A quadratic field which is Euclidean but not norm-Euclidean
  | journal=Manuscripta Mathematica
  | volume=83
  | pages=327–330
  | doi=10.1007/BF02567617
 | url=http://www.math.clemson.edu/~jimlb/CourseNotes/AbstractAlgebra/EuclideanNotNormEuclidean.pdf
  | archive-url=https://web.archive.org/web/20150129163358/http://www.math.clemson.edu/~jimlb/CourseNotes/AbstractAlgebra/EuclideanNotNormEuclidean.pdf
  | archive-date=2015-01-29
  }}
* {{citation
  | last1=de Bruijn | first1=N. G.
  | year=1981
  | title=Algebraic theory of Penrose's non-periodic tilings of the plane, I, II
  | journal=Indagationes Mathematicae
  | volume=43
  | issue=1
  | pages=39–66
  | url=http://alexandria.tue.nl/repository/freearticles/597566.pdf
  }}
* {{citation
  | last1=Dedekind | first1=Richard | author1-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=2nd
  | year=1871
  | publisher=Vieweg
  | access-date=2009-08-05 
  }}
* {{citation
  | last1=Dummit | first1=D. S.
  | last2=Foote | first2=R. M.
  | year=2004
  | title=Abstract Algebra
  | edition=3rd
  }}
* {{citation
  | last1=Harper |first1=M.
  | year=2004
  | title=<math>\mathbb {Z}[\sqrt{14}]</math> ''is Euclidean''
  | journal=Can. J. Math. 56
  |volume=56
 | pages=55–70
  |doi=10.4153/CJM-2004-003-9
 }}
* {{cite book
  | last1 = LeVeque | first1 = William J. | author1-link = William J. LeVeque
  | title = Topics in Number Theory, Volumes I and II
  | publisher = Dover Publications | location = New York
  | year = 2002 | orig-year = 1956
  | isbn = 978-0-486-42539-9
  | zbl = 1009.11001
  | url = https://archive.org/details/topicsinnumberth0000leve
  }}
{{refend}}

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

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