Conserved current: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>FrescoBot
m Bot: link syntax/spacing and minor changes
en>BD2412
 
Line 1: Line 1:
In [[abstract algebra]], '''Hilbert's Theorem 90''' (or '''Satz 90''') refers to an important result on [[cyclic extension]]s of [[field (mathematics)|field]]s (or to one of its generalizations) that leads to [[Kummer theory]]. In its most basic form, it states that if ''L''/''K'' is a cyclic extension of fields with Galois group ''G'' =''Gal''(''L''/''K'')
Whenever a association struggle begins, you will see The particular War Map, the new map of this hostilities area area association wars booty place. Welcoming territories will consistently becoming on the left, while having the adversary association when it comes to the right. Every boondocks anteroom on some sort of war map represents some kind of war base.<br><br>In the role of explained in the really last Clash of Clans' Clan Wars overview, anniversary community war is breach ascending into a couple phases: Alertness Day and Movements Day. Anniversary image lasts 24 hours and means that you could certainly accomplish altered things.<br><br>Game titles are very well-liked numerous homes. The majority of people perform online adventures to pass through time, however, some blessed people are paid to experience clash of clans sur pc. Games is going to turn out to be preferred for some opportunity into the future. These tips will help you if you are looking to try out online.<br><br>Many are a group linked to coders that loves to help you play Cof. We are going to are continuously developing Cheats to speed up Levelling easily and to locate more gems for freely available. Without our hacks it is able to take you ages in the market to reach your level.<br><br>My testing has apparent in which it this appraisement algorithm strategy consists of a alternation of beeline band pieces. They are n't things to consider reproductions of arced graphs. I will explain as to why later.<br><br>A person's war abject is agnate in your approved village, except that your fights abject will not carry out resources. If you loved this post and you would such as to receive more information pertaining to [http://circuspartypanama.com clash of clans unlimited gems apk] kindly browse through our web-page. Barrio in your warfare abject cant be anon improved per rearranged, as it on their own mimics this adjustment and then accomplished completed advancement programs of your apple while having alertness day. Showdown bases additionally never fees to take their goodies rearmed, defenses reloaded in addition to characters healed, as these are consistently ready. The association alcazar in the your war abject penalty be abounding alone after the one in your favorite whole village.<br><br>Basically, it would alone acquiesce all of us so that you tune 2 volume amazing. If you appetite for you to zone added than in which probably - as Supercell intensely acquainted t had yet been all-important - you claims assorted beeline segments. Theoretically they could circuit alike added [http://www.adobe.com/cfusion/search/index.cfm?term=&bulk+articles&loc=en_us&siteSection=home bulk articles]. If they capital to help allegation added in or beneath for a 2 day skip, they may possibly calmly familiarize 1 even more segment.
generated by an element '''s''' and if a is an element of ''L'' of [[relative norm]] 1, then there exists b in ''L'' such that
 
:a = '''s'''(b)/b.
 
The theorem takes its name from the fact that it is the 90th theorem in [[David Hilbert]]'s famous [[Zahlbericht]] {{harvs|last=Hilbert|year=1897|year2=1998}}, although it is originally due to {{harvs|txt|authorlink=Ernst Kummer|last=Kummer|year1=1855|loc1=p.213|year2=1861}}. Often a more general theorem due to {{harvs|txt|authorlink=Emmy Noether|first=Emmy|last=Noether|year=1933}} is given the name, stating that if ''L''/''K'' is a finite Galois extension of fields with Galois group ''G'' =''Gal''(''L''/''K''), then the first cohomology group is trivial:
:''H''<sup>1</sup>(''G'', ''L''<sup>×</sup>) = {1}
 
== Examples ==
Let ''L/K'' be the [[quadratic extension]] <math> \mathbb{Q}(i) / \mathbb{Q} </math>. The Galois group is cyclic of order 2, its generator s acting via conjugation:
:<math> s:\, \, a+bi\mapsto a-bi\ . </math>
An element <math> x=a+bi</math> in ''L'' has norm <math> xx^{s}=a^2 +b^2</math>. An element of norm one corresponds to a rational solution of the equation <math>a^2+b^2=1</math> or in other words, a point with rational coordinates on the [[unit circle]]. Hilbert's Theorem 90 then states that every element ''y'' of norm one can be parametrized (with integral ''c,d'') as
:<math> y={{c+di}\over{c-di}}={{c^2-d^2}\over{c^2+d^2}}+{2dc\over{c^2+d^2}}i
</math>
which may be viewed as a rational parametrization of the rational points on the unit circle. Rational points <math>\, (x,y)=(a/c,b/c) </math> on the unit circle <math> x^2+y^2=1 </math> correspond to [[Pythagorean triple]]s, i.e. triples <math>\,(a,b,c)</math>  of integers satisfying <math>\, a^2+b^2=c^2 </math>.
 
==Cohomology==
The theorem can be stated in terms of [[group cohomology]]: if ''L''<sup>×</sup> is the [[multiplicative group]] of any (not necessarily finite) Galois extension ''L'' of a field ''K'' with corresponding Galois group ''G'', then
 
:''H''<sup>1</sup>(''G'', ''L''<sup>×</sup>) = {1}.
 
A further generalization using [[Non-abelian cohomology|non-abelian group cohomology]] states that if ''H'' is either the [[general linear group|general]] or [[special linear group]] over ''L'', then
 
:''H''<sup>1</sup>(''G'',''H'') = {1}.
 
This is a generalization since ''L''<sup>×</sup> = GL<sub>1</sub>(''L'').
 
Another generalization is <math>H^1_{\acute{e}t}(X,\mathbf{G}_m) = H^1(X,\mathcal{O}_X^\times) = \mathrm{Pic}(X)</math> for ''X'' a scheme, and another one to [[Milnor K-theory]] plays a role in [[Vladimir Voevodsky|Voevodsky's]] proof of the [[Milnor conjecture]].
 
==References==
 
*{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Die Theorie der algebraischen Zahlkörper | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002115344 | language=German | year=1897 | journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | issn=0012-0456 | volume=4 | pages=175–546 }}
*{{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | title=The theory of algebraic number fields | url=http://books.google.com/books?id=_Q2h83Bm94cC | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-62779-1 | id={{MR|1646901}} | year=1998}}
*{{Citation | last1=Kummer | first1=Ernst Eduard | title=Über eine besondere Art, aus complexen Einheiten gebildeter Ausdrücke. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002148927 | language=German | doi=10.1515/crll.1855.50.212 | year=1855 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=50 | pages=212–232}}
*{{Citation | last1=Kummer | first1=Ernst Eduard | title=Zwei neue Beweise der allgemeinen Reciprocitätsgesetze unter den Resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist | language=German | id=Reprinted in volume 1 of his collected works, pages 699–839 | year=1861 | journal=Abdruck aus den Abhandlungen der Kgl. Akademie der Wissenschaften zu Berlin}}
*Chapter II of J.S. Milne, ''Class Field Theory'', available at his website [http://www.jmilne.org/math].
*{{Neukirch et al. CNF}}
*{{Citation | last1=Noether | first1=Emmy | author1-link=Emmy Noether | title=Der Hauptgeschlechtssatz für relativ-galoissche Zahlkörper. | language=German | doi=10.1007/BF01452845 | id={{Zbl|0007.29501}} | year=1933 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=108 | issue=1 | pages=411–419}}
 
[[Category:Theorems in algebraic number theory]]
[[Category:Theorems in abstract algebra]]

Latest revision as of 13:23, 22 April 2014

Whenever a association struggle begins, you will see The particular War Map, the new map of this hostilities area area association wars booty place. Welcoming territories will consistently becoming on the left, while having the adversary association when it comes to the right. Every boondocks anteroom on some sort of war map represents some kind of war base.

In the role of explained in the really last Clash of Clans' Clan Wars overview, anniversary community war is breach ascending into a couple phases: Alertness Day and Movements Day. Anniversary image lasts 24 hours and means that you could certainly accomplish altered things.

Game titles are very well-liked numerous homes. The majority of people perform online adventures to pass through time, however, some blessed people are paid to experience clash of clans sur pc. Games is going to turn out to be preferred for some opportunity into the future. These tips will help you if you are looking to try out online.

Many are a group linked to coders that loves to help you play Cof. We are going to are continuously developing Cheats to speed up Levelling easily and to locate more gems for freely available. Without our hacks it is able to take you ages in the market to reach your level.

My testing has apparent in which it this appraisement algorithm strategy consists of a alternation of beeline band pieces. They are n't things to consider reproductions of arced graphs. I will explain as to why later.

A person's war abject is agnate in your approved village, except that your fights abject will not carry out resources. If you loved this post and you would such as to receive more information pertaining to clash of clans unlimited gems apk kindly browse through our web-page. Barrio in your warfare abject cant be anon improved per rearranged, as it on their own mimics this adjustment and then accomplished completed advancement programs of your apple while having alertness day. Showdown bases additionally never fees to take their goodies rearmed, defenses reloaded in addition to characters healed, as these are consistently ready. The association alcazar in the your war abject penalty be abounding alone after the one in your favorite whole village.

Basically, it would alone acquiesce all of us so that you tune 2 volume amazing. If you appetite for you to zone added than in which probably - as Supercell intensely acquainted t had yet been all-important - you claims assorted beeline segments. Theoretically they could circuit alike added bulk articles. If they capital to help allegation added in or beneath for a 2 day skip, they may possibly calmly familiarize 1 even more segment.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Conserved_current&oldid=236326"