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In [[mathematics]], the '''ring of integers''' of an [[algebraic number field]]&nbsp;{{mvar|K}} is the [[ring (mathematics)|ring]] of all [[integral element]]s contained in&nbsp;{{mvar|K}}. An integral element is a [[root of a polynomial|root of]] a [[monic polynomial]] with [[rational integer]] coefficients, {{math|size=120%|''x''<sup>''n''</sup> + ''c''<sub>''n''−1</sub>''x''<sup>''n''−1</sup> + … + ''c''<sub>0</sub> }}. This ring is often denoted by {{math|size=120%|O<sub>''K''</sub>}} or <math>\mathcal O_K</math>. Since any [[integer]] number belongs to&nbsp;{{mvar|K}} and is its integral element, the ring&nbsp;{{math|'''Z'''}} is always a [[subring]] of&nbsp;{{math|size=120%|O<sub>''K''</sub>}}.
 
The ring&nbsp;{{math|'''Z'''}} is the simplest possible ring of integers.<ref>''The ring of integers'', without specifying the field, refers to the ring&nbsp;{{math|'''Z'''}} of "ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "[[wikt:integer|integer]]" in abstract algebra.</ref> Namely, {{math|size=120%|1='''Z''' = O<sub>'''Q'''</sub>}} where {{math|'''Q'''}} is the [[Field (mathematics)|field]] of [[rational number]]s. And indeed, in [[algebraic number theory]] the elements of&nbsp;{{math|'''Z'''}} are often called the "rational integers" because of this.
 
The ring of integers  of a number field is the unique maximal [[order (ring theory)|order]] in the field.
 
== Properties ==
The ring of integers {{math|size=120%|O<sub>''K''</sub>}} is a {{math|'''Z'''}}-[[module (mathematics)|module]]. Indeed it is a [[free module|free]] {{math|'''Z'''}}-module, and thus has an '''integral basis''', that is a [[basis (linear algebra)|basis]] {{math|size=120%|''b''<sub>1</sub>, … ,''b''<sub>''n''</sub> ∈ O<sub>''K''</sub>}} of the {{math|'''Q'''}}-vector&nbsp;space&nbsp;{{mvar|K}} such that each element&nbsp;{{mvar|x}} in {{math|size=120%|O<sub>''K''</sub>}} can be uniquely represented as
:<math>x=\sum_{i=1}^na_ib_i,</math>
with {{math|size=120%|''a''<sub>''i''</sub> ∈ '''Z'''}}. The rank&nbsp;{{mvar|n}} of {{math|size=120%|O<sub>''K''</sub>}} as a free {{math|'''Z'''}}-module is equal to the [[Degree of a field extension|degree]] of&nbsp;{{mvar|K}} over {{math|'''Q'''}}.
 
The rings of integers in number fields are [[Dedekind domain]]s.
 
==Examples==
If {{mvar|p}} is a [[prime number|prime]], ζ is a {{mvar|p}}th [[root of unity]] and {{math|size=120%|1=''K'' = '''Q'''(ζ)}} is the corresponding [[cyclotomic field]], then an integral basis of {{math|size=120%|1=O<sub>''K''</sub> = '''Z'''[ζ]}} is given by {{math|size=120%|(1, ζ, ζ<sup>2</sup>, … , ζ<sup>''p''−2</sup>)}}.
 
If {{mvar|d}} is a [[square-free integer]] and {{math|size=120%|1=''K'' = '''Q'''({{sqrt|''d''}})}} is the corresponding [[quadratic field]], then {{math|size=120%|O<sub>''K''</sub>}} is a ring of [[quadratic integer]]s and its integral basis is given by {{math|size=120%|(1, (1 + {{sqrt|''d''}})/2)}} if {{math|''d'' ≡ 1 ([[Modular arithmetic|mod]] 4)}} and by {{math|size=120%|(1, {{sqrt|''d''}})}} if {{math|''d'' ≡ 2, 3 (mod 4)}}.
 
==Generalization==
One defines the ring of integers of a [[non-archimedean local field]] F as the set of all elements of F with absolute value ≤1; this is a ring because of the strong triangle inequality. If F is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers.
 
For example, the {{mvar|p}}-adic integers {{math|size=120%|'''Z'''<sub>''p''</sub>}} are the ring of integers of the [[p-adic number|{{mvar|p}}-adic numbers]] {{math|size=120%|'''Q'''<sub>''p''</sub> }}.
 
==References==
*{{Neukirch ANT}}
 
== Notes ==
{{reflist}}
 
[[Category:Ring theory]]
[[Category:Algebraic number theory]]

Latest revision as of 04:54, 31 December 2014

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