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In [[mathematics]], '''Dirichlet's unit theorem''' is a basic result in [[algebraic number theory]] due to [[Peter Gustav Lejeune Dirichlet]].<ref>{{harvnb|Elstrodt|2007|loc=§8.D}}</ref> It determines the [[rank of an abelian group|rank]] of the [[group of units]] in the [[ring (mathematics)|ring]] ''O''<sub>''K''</sub> of [[algebraic integer]]s of a [[number field]] ''K''. The '''regulator''' is a positive real number that determines how "dense" the units are.
== 蘭大人は非常に不吉なああ見て ==


==Dirichlet's unit theorem==
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The statement is that the group of units is finitely generated and has [[Rank of an abelian group|rank]] (maximal number of multiplicatively independent elements) equal to
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:''r'' = ''r''<sub>1</sub> + ''r''<sub>2</sub> &minus; 1
== 9 4 8ヘッドフォン ==


where ''r''<sub>1</sub> is the ''number of real embeddings'' and ''r''<sub>2</sub> the ''number of conjugate pairs of complex embeddings'' of ''K''. This characterisation of
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''r''<sub>1</sub> and ''r''<sub>2</sub> is based on the idea that there will be as many ways to embed ''K'' in the [[complex number]] field as the degree ''n'' = [''K'' : '''Q''']; these will either be into the [[real number]]s, or pairs of embeddings related by [[complex conjugation]], so that
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</ul>


:''n'' = ''r''<sub>1</sub> + 2''r''<sub>2</sub>.
== Jiang Lan said that the two methods are feasible. ==


Note that if ''K'' is Galois over ''Q'' then either ''r''<sub>1</sub> is non-zero or ''r''<sub>2</sub> is non-zero, but not both.
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Other ways of determining ''r''<sub>1</sub> and ''r''<sub>2</sub> are
== whistling wind ==


* use the [[Primitive element (field theory)|primitive element]] theorem to write ''K'' = '''Q'''(α), and then ''r''<sub>1</sub> is the number of [[conjugate element (field theory)|conjugates]] of α that are real, 2''r''<sub>2</sub> the number that are complex;
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* write the [[tensor product of fields]] ''K'' ⊗<sub>'''Q'''</sub>'''R''' as a product of fields, there being ''r''<sub>1</sub> copies of '''R''' and ''r''<sub>2</sub> copies of '''C'''.
<ul>
 
 
As an example, if ''K'' is a [[quadratic field]], the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field. The theory for real quadratic fields is essentially the theory of [[Pell's equation]].
  <li>[http://www.regalmain.com/bbs/forum.php?mod=viewthread&tid=9732 http://www.regalmain.com/bbs/forum.php?mod=viewthread&tid=9732]</li>
 
 
The rank is > 0 for all number fields besides '''Q''' and imaginary quadratic fields, which have rank 0. The 'size' of the units is measured in general by a [[determinant]] called the regulator. In principle a basis for the units can be effectively computed; in practice the calculations are quite involved when ''n'' is large.
  <li>[http://www.qtfabiao.com/plus/feedback.php?aid=7 http://www.qtfabiao.com/plus/feedback.php?aid=7]</li>
 
 
The torsion in the group of units is the set of all roots of unity of ''K'', which form a finite [[cyclic group]]. For a number field with at least one real embedding the torsion
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must therefore be only {1,&minus;1}.  There are number fields, for example most [[imaginary quadratic field]]s, having no real embeddings which also have {1,&minus;1} for the torsion of its unit group.
    
 
  </ul>
Totally real fields are special with respect to units. If ''L/K'' is a finite extension of number fields with degree greater than 1 and
the units groups for the integers of ''L'' and ''K'' have the same rank then ''K'' is totally real and ''L'' is a totally complex quadratic extension. The converse
holds too. (An example is
''K'' equal to the rationals and ''L'' equal to an imaginary quadratic field; both have unit rank 0.)
 
There is a generalisation of the unit theorem by [[Helmut Hasse]] (and later [[Claude Chevalley]]) to describe the structure of the group of ''[[S-unit]]s'', determining the rank of the unit group in [[localization of a ring|localizations]] of rings of integers. Also, the [[Galois module]] structure of <math>\mathbf{Q} \oplus O_{K,S} \otimes_\mathbf{Z} \mathbf{Q}</math> has been determined.{{sfn|Neukirch|Schmidt|Wingberg|2000|loc=proposition VIII.8.6.11}}
 
==The regulator==
 
Suppose that ''u''<sub>1</sub>,...,''u''<sub>r</sub> are a set of generators for the unit group modulo roots of unity. If ''u'' is an algebraic number, write ''u''<sup>1</sup>, ..., ''u''<sup>''r+1''</sup> for the different embeddings into '''R''' or '''C''', and set
''N''<sub>''j''</sub> to 1, resp. 2 if corresponding embedding is real, resp. complex.
Then the ''r'' by ''r''&nbsp;+&nbsp;1 matrix whose entries are <math>N_j\log|u_i^j|</math> has the property that the sum of any row is zero (because all units have norm 1, and the log of the norm is the sum of the entries of a row). This implies that the absolute value ''R'' of the determinant of the submatrix formed by deleting one column is independent of the column.
The number ''R'' is called the '''regulator''' of the algebraic number field (it does not depend on the choice of generators ''u''<sub>i</sub>). It measures the "density" of the units: if the regulator is small, this means that there are "lots" of units.
 
The regulator has the following geometric interpretation. The map taking a unit ''u'' to the vector with entries <math>N_j\log|u^j|</math> has image in the ''r''-dimensional subspace of '''R'''<sup>''r''+1</sup> consisting
of all vector whose entries have sum 0, and by Dirichlet's unit theorem the image is a lattice in this subspace. The volume of a fundamental domain of this lattice is  ''R''√(''r''+1).
 
The regulator of an algebraic number field of degree greater than 2 is usually quite cumbersome to calculate, though there are now computer algebra packages that can do it in many cases. It is usually much easier to calculate the product ''hR'' of the [[class number (number theory)|class number]] ''h'' and the regulator using the [[class number formula]], and the main difficulty in calculating the class number of an algebraic number field is usually the calculation of the regulator.
 
===Examples===
[[Image:Discriminant49CubicFieldFundamentalDomainOfUnits.png|thumb|300px|right|A fundamental domain in logarithmic space of the group of units of the cyclic cubic field ''K'' obtained by adjoining to '''Q''' a root of ''f''(''x'')&nbsp;=&nbsp;''x''<sup>3</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;&minus;&nbsp;2''x''&nbsp;&minus;&nbsp;1. If &alpha; denotes a root of ''f''(''x''), then a set of fundamental units is {ε<sub>1</sub>,&nbsp;ε<sub>2</sub>} where ε<sub>1</sub>&nbsp;=&nbsp;α<sup>2</sup>&nbsp;+&nbsp;α&nbsp;&minus;&nbsp;1 and ε<sub>2</sub>&nbsp;=&nbsp;2&nbsp;&minus;&nbsp;α<sup>2</sup>. The area of the fundamental domain is approximately 0.910114, so the regulator of ''K'' is approximately 0.525455.]]
*The regulator of an [[imaginary quadratic field]], or of the rational integers, is 1 (as the determinant of a 0&times;0 matrix is 1).
*The regulator of a [[real quadratic field]] is the logarithm of its [[fundamental unit]]: for example, that of '''Q'''(√5) is log((√5&nbsp;+&nbsp;1)/2). This can be seen as follows. A fundamental unit is (√5 + 1)/2, and its images under the two embeddings into '''R''' are (√5&nbsp;+&nbsp;1)/2 and (&minus;√5&nbsp;+&nbsp;1)/2. So the ''r'' by ''r''&nbsp;+&nbsp;1 matrix is
 
::<math>\left[1\times\log\left|{\sqrt{5} + 1 \over 2}\right|, \quad 1\times \log\left|{-\sqrt{5} + 1 \over 2}\right|\ \right].</math>
 
*The regulator of the [[cyclic cubic field]] '''Q'''(α), where α is a root of ''x''<sup>3</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;&minus;&nbsp;2''x''&nbsp;&minus;&nbsp;1, is approximately 0.5255. A basis of the group of units modulo roots of unity is {ε<sub>1</sub>,&nbsp;ε<sub>2</sub>} where ε<sub>1</sub>&nbsp;=&nbsp;α<sup>2</sup>&nbsp;+&nbsp;α&nbsp;&minus;&nbsp;1 and ε<sub>2</sub>&nbsp;=&nbsp;2&nbsp;&minus;&nbsp;α<sup>2</sup>.<ref>{{harvnb|Cohen|1993|loc=Table B.4}}</ref>
 
==Higher regulators==
 
A 'higher' regulator refers to a construction for a function on an [[algebraic K-group]] with index ''n'' > 1 that plays the same role as the classical regulator does for the group of units, which is a group ''K''<sub>1</sub>. A theory of such regulators has been in development, with work of [[Armand Borel]] and others. Such higher regulators play a role, for example, in the [[Beilinson conjectures]], and are expected to occur in evaluations of certain [[L-function]]s at integer values of the argument.<ref name=Bloch>{{cite book | last=Bloch | first=Spencer J. | authorlink=Spencer Bloch | title=Higher regulators, algebraic ''K''-theory, and zeta functions of elliptic curves | series=CRM Monograph Series | volume=11 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2000 | isbn=0-8218-2114-8 | zbl=0958.19001 }}</ref>
 
==Stark regulator==
The formulation of [[Stark's conjectures]] led [[Harold Stark]] to define what is now called the '''Stark regulator''', similar to the classical regulator as a determinant of logarithms of units, attached to any [[Artin representation]].<ref>[http://www.math.tifr.res.in/~dprasad/artin.pdf PDF]</ref><ref>[http://www.math.harvard.edu/~dasgupta/papers/Dasguptaseniorthesis.pdf PDF]</ref>
 
==''p''-adic regulator==
Let ''K'' be a [[number field]] and for each [[Valuation (algebra)|prime]] ''P'' of ''K'' above some fixed rational prime ''p'', let ''U''<sub>''P''</sub> denote the local units at ''P'' and let ''U''<sub>1,''P''</sub> denote the subgroup of principal units in ''U''<sub>''P''</sub>. Set
 
: <math> U_1 = \prod_{P|p} U_{1,P}. </math>
 
Then let ''E''<sub>1</sub> denote the set of global units ''ε'' that map to ''U''<sub>1</sub> via the diagonal embedding of the global units in&nbsp;''E''.
 
Since <math>E_1</math> is a finite-[[Index of a subgroup|index]] subgroup of the global units, it is an [[abelian group]] of rank <math>r_1 + r_2 - 1</math>.  The '''''p''-adic regulator''' is the determinant of the matrix formed by the ''p''-adic logarithms of the generators of this group.  ''[[Leopoldt's conjecture]]'' states that this determinant is non-zero.<ref name=NSW6267>Neukirch et al (2008) p.626-627</ref><ref>{{cite book | last=Iwasawa | first=Kenkichi | authorlink=Kenkichi Iwasawa | title=Lectures on ''p''-adic ''L''-functions | series=Annals of Mathematics Studies | volume=74 | location=Princeton, NJ | publisher=Princeton University Press and University of Tokyo Press | year=1972 | isbn=0-691-08112-3 | zbl=0236.12001 | pages=36-42 }}</ref>
 
==See also==
*[[Elliptic unit]]
*[[Cyclotomic unit]]
*[[Shintani's unit theorem]]
 
==Notes==
{{Reflist}}
 
==References==
* {{Cite book
| last=Cohen
| first=Henri
| author-link=Henri Cohen (number theorist)
| title=A Course in Computational Algebraic Number Theory
| publisher=[[Springer-Verlag]]
| location=Berlin, New York
| series=[[Graduate Texts in Mathematics]] | volume=138
| isbn=978-3-540-55640-4
| mr=1228206 | zbl=0786.11071
| year=1993
| ref=harv
}}
*{{cite journal | last = Elstrodt | first = Jürgen | journal = Clay Mathematics Proceedings
  | title = The Life and Work of Gustav Lejeune Dirichlet (1805–1859) | work = | publisher = | year = 2007
  | url = http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf | format = PDF | doi =
   | accessdate = 2010-06-13 | ref = harv}}
* {{cite book | first=Serge | last=Lang | author-link=Serge Lang | title=Algebraic number theory | edition=2nd | series=Graduate Texts in Mathematics | volume=110 | location=New York | publisher=[[Springer-Verlag]] | year=1994 | isbn=0-387-94225-4 | zbl=0811.11001 }}
*{{Neukirch ANT}}
*{{Neukirch et al. CNF}}
 
[[Category:Theorems in algebraic number theory]]

Revision as of 08:17, 26 February 2014

蘭大人は非常に不吉なああ見て

あなたの資格が非常に良いではありません。蘭大人は非常に不吉なああ見て。 '
彼は秦ゆうの本体に行く前
シルバーフラワー祖母の話 クリスチャンルブタンジャパン。

秦Yuは肩を撫で、シルバーフラワー親切に祖母は言った：あなたはとても驚くほど練習スピードだ」の後にあなたが持続されている場合は、私は数万年、あなたはその後、驚くべき資格にあなたに神の領域に達することができるようになります......参照プラス蘭の大人のヘルプは、またはであること、そして子どもたちに一緒に立つことがあります。 クリスチャンルブタン メンズ 通販 '

秦Yuはうなずいた クリスチャンルブタン 店舗 東京。

「素晴らしい資質が、私はまだあなたが広場でハイエンドに認定されていますか見てなかったのか？ 'シルバーフラワー祖母の心違法チャンネルを、「どのようにちょうど3000​​年の間、魂は、この手順を実行するだろうか？'

秦ゆう今回チャネル：「おばあちゃんが言った、秦ゆうが私の心には決して忘れない、私は知らない何秦ゆうを思い出させるおばあちゃん。 靴 クリスチャンルブタン '
シルバーフラワー祖母への大きなスタンドを持つ子どものために
、秦Yuは非常に丁重にまだある。

シルバーフラワー祖母が突然言った： クリスチャンルブタン通販 'それは思い出させる」ことになると、私は非常に重要なことだと思います。「シルバーフラワー祖母はヤンを見て 相关的主题文章：

9 4 8ヘッドフォン

太い根鋭いスパイク、額だけでなく、黄金の角を持っています。

瞬間竜岩秦ゆう今まで見たライオン、マリーXiongshou一般のようなもので、4速スプリントは雷 -

でもトップグレードfeijian、数メートルの厚さの竜岩ライオン竜岩の上に刺すことができるように期待することはできません。 「ランブル〜〜〜 クリスチャンルブタン 直営店 '彼はfeijianフラグメンテーションを見すぎて、竜岩ライオンハウリング、ゴールデンホーン恐ろしい光を点滅して、暗赤色のぼかしは、飛んで頭蓋骨、骨折した足の切り株を投げ、何度も何度も悲鳴空、血液ドリフトのフルひき肉......

オクタゴン中国​​語 - 無料のVIPフィクション

」は、動物は動物で、逃げる。 '
法執行機関の
グリーンドラゴンの宮殿洞窟虚レルムは大声で、現時点では険しい顔を叫んだが、一部は怯えた目である クリスチャンルブタン 値段。

クリスチャンルブタン 銀座......
ボリューム上
、竜岩ライオンより1テロリストが存在する - クリスチャンルブタン 日本 正確ハイドラ、、それは9龍」と呼ぶことができる、100メートル以上のヒドラ量、そして最も重要なのは、彼の頭のすべての9長い10メートルを持っています。

9 4 クリスチャンルブタン 銀座 8ヘッドフォン 相关的主题文章：

Jiang Lan said that the two methods are feasible.

Tears. You have a drop. Once the two of you alike. Combination of the two of you to comprehend. The two of you. Some people will easily become the king of gods. '

'And Tears drops クリスチャンルブタン 日本 of life of the soul is the life of God king fall. peculiar to the energy of the body, the tear drops of life soul fusion energy had life God King will appear again. Whether it drops the soul of life Tears to whom, and who are likely to become a new 'life Gods'. '

Jiang Lan said this. Qin Yu heart also some understand.

meteor tears itself. Is composed of two parts. One is the perception of space law. Another is the life energy of God, the king, the king of the energy of the クリスチャンルブタン 東京 life of God is a very peculiar クリスチャンルブタン 銀座 way, and even allows the soul to repair.

'with you. stand children she is クリスチャンルブタン 偽物 likely to become the king of gods, their daughter became the king of the gods. This ginger Vatican should also have great appeal.' said Jiang Lan クリスチャンルブタン 店舗 smile.

Qin Yu nodded.

right, Jiang Lan said that the two methods are feasible. 相关的主题文章：

whistling wind

France became the innate クリスチャンルブタン セール master ah. According to Yun Xing expected, today's Kohane external work to repair it is estimated that only around mid day after tomorrow. 'QIN heart Antan.

internal strength expert, outsiders can judge its repair is based on internal forces. But experts external work is to comprehension, but also unable to determine how strong the other クリスチャンルブタン ブーツ muscles in the end. After all, the strength of muscle strength, it is クリスチャンルブタン バッグ difficult to judge it, how many pounds can only use one hand to judge it.

QIN mind, his three sons Qin Yu is just a penance external work, but in the end it may not have a child.

'and so find feather, let him come to me. クリスチャンルブタン バッグ As stone Yan Chi refining iron in the end what クリスチャンルブタン 店舗 night we re-negotiate.' QIN finished, turn left directly, at the moment the mood is a bit messy QIN the ......

twenty-second chapter medal task (on)

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night, Qin Yu, sitting cross-legged Hawk back, Blackhawk is flying among the clouds.

whistling wind, like a knife pierced in general face Qin Yu 相关的主题文章：

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