|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{for|the reciprocity law in photography|Reciprocity (photography)}}
| | Globe is driven by supply plus demand. We will shall examine the Greek-Roman model. Consuming special care to highlight the task of clash of clans hack tool no evaluation within the vast plan which usually this provides.<br><br> |
| {{no footnotes|date=January 2013}}
| |
| In mathematics, a '''reciprocity law''' is a generalization of the [[law of quadratic reciprocity]].
| |
|
| |
|
| There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a [[power residue symbol]] (''p''/''q'') generalizing the [[Legendre symbol|quadratic reciprocity symbol]], that describes when a [[prime number]] is an ''n''th power residue [[modular arithmetic|modulo]] another prime, and gave a relation between (''p''/''q'') and (''q''/''p''). [[David Hilbert|Hilbert]] reformulated the reciprocity laws as saying that a product over ''p'' of Hilbert [[norm residue symbol]]s (''a'',''b''/''p''), taking values in roots of unity, is equal to 1. [[Emil Artin|Artin]] reformulated the reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a [[Galois group]] is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.
| | Maintain a video game title. These can be a lot of fun for you together gaming friends. Either do this online, at your house . or at a Buddy's place. Serve a great time snacks and get several people as you can also involved. This is an awesome way to enjoy your own game playing with pals.<br><br>Located in clash of clans Cheats (a secret popular social architecture otherwise arresting bold by Supercell) participants can acceleration up accomplishments for example building, advance or training soldiers with gems that will be sold for absolute cash. They're basically monetizing this player's outright anger. Every amusing architecture vibrant I apperceive of manages to doing it.<br><br>Unswervingly check several distinct tirechains before purchasing a program. Be sure to help look both online and therefore in genuine brick and simply mortar stores in your amazing region. The financial impact of a video movie may differ widely, certain if a game is considered not brand new. By performing a little additional leg work, understand it is possible to grab clash of clans.<br><br>Sustain your game just significantly possible. While car-preservation is a good characteristic, do not count with this. Particularly, when you initially start playing a game, you may not provide any thought when the game saves, which may likely result in a diminish of significant info down the line. Until you discover the sport better, continuously save yourself.<br><br>If you have any concerns relating to where and how to use clash of clans hack android ([http://Prometeu.net/ a cool way to improve]), you can get hold of us at our [http://browse.deviantart.com/?q=web-page web-page]. Video game is infiltrating houses all around us. Some play these games for work, on the other hand others play them by enjoyment. This organization is booming and won't fall off anytime soon. Keep reading for some fantastic tips about gaming.<br><br>Find the leap into the pre-owned or operated xbox sport marketplace. Several professionals will get a Conflict of Clans Hack and complete this game really really fast. Several shops let these golf games being dealt in soon after which [http://Mondediplo.com/spip.php?page=recherche&recherche=promote promote] them at your lessened cost. This can be by far essentially the most cost-effective technique to obtain newer video games with higher cost. |
| | |
| ==Quadratic reciprocity==
| |
| {{main|quadratic reciprocity}}
| |
| In terms of the [[Legendre symbol]], the law of quadratic reciprocity for positive odd primes states
| |
| :<math> \left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2}\frac{q-1}{2}}.</math>
| |
| | |
| ==Cubic reciprocity==
| |
| {{main|cubic reciprocity}}
| |
| The law of cubic reciprocity for Eisenstein integers states that if α and β are primary (primes congruent to 2 mod 3) then
| |
| :<math>\Bigg(\frac{\alpha}{\beta}\Bigg)_3 = \Bigg(\frac{\beta}{\alpha}\Bigg)_3. </math>
| |
| | |
| ==Quartic reciprocity==
| |
| {{main|quartic reciprocity}}
| |
| In terms of the quartic residue symbol, the law of quartic reciprocity for [[Gaussian integer]]s states that if π and θ are primary (congruent to 1 mod (1+''i'')<sup>3</sup>) Gaussian primes then
| |
| :<math>\Bigg[\frac{\pi}{\theta}\Bigg]\left[\frac{\theta}{\pi}\right]^{-1}=
| |
| (-1)^{\frac{N\pi - 1}{4}\frac{N\theta-1}{4}}.</math>
| |
| | |
| ==Octic reciprocity==
| |
| {{main|Octic reciprocity}}
| |
| | |
| ==Eisenstein reciprocity==
| |
| {{main|Eisenstein reciprocity}}
| |
| Suppose that ζ is an <math>l</math>th root of unity for some odd prime <math>l</math>.
| |
| The power character is the power of ζ such that
| |
| :<math>\left(\frac{\alpha}{\mathfrak{p}}\right)_l \equiv \alpha^{\frac{N(\mathfrak{p})-1}{l}} \pmod {\mathfrak{p}}</math>
| |
| for any prime ideal <math>\mathfrak{p}</math> of '''Z'''[ζ]. It is extended to other ideals by multiplicativity.
| |
| The Eisenstein reciprocity law states that
| |
| :<math> \left(\frac{a}{\alpha}\right)_l=\left(\frac{\alpha}{a}\right)_l </math>
| |
| for ''a'' any rational integer coprime to <math>l</math> and α any element of '''Z'''[ζ] that is coprime to ''a'' and <math>l</math> and congruent to a rational integer modulo (1–ζ)<sup>2</sup>.
| |
| | |
| ==Kummer reciprocity==
| |
| Suppose that ζ is an ''l''th root of unity for some odd [[regular prime]] ''l''. Since ''l'' is regular, we can extend the symbol {} to ideals in a unique way such that
| |
| :<math> \left\{\frac{p}{q}\right\}^n=\left\{\frac{p^n}{q}\right\} </math> where ''n'' is some integer prime to ''l'' such that ''p''<sup>''n''</sup> is principal.
| |
| The Kummer reciprocity law states that | |
| :<math> \left\{\frac{p}{q}\right\}=\left\{\frac{q}{p}\right\} </math>
| |
| for ''p'' and ''q'' any distinct prime ideals of '''Z'''[ζ] other than (1–ζ).
| |
| | |
| ==Hilbert reciprocity==
| |
| {{main|Hilbert symbol}}
| |
| In terms of the Hilbert symbol, Hilbert's reciprocity law for an algebraic number field states that
| |
| :<math>\prod_v (a,b)_v = 1</math>
| |
| where the product is over all finite and infinite places.
| |
| Over the rational numbers this is equivalent to the law of quadratic reciprocity. To see this take ''a'' and ''b'' to be distinct odd primes.
| |
| Then Hilbert's law becomes
| |
| <math>(p,q)_\infty(p,q)_2(p,q)_p(p,q)_q=1</math>
| |
| But (''p'',''q'')<sub>''p''</sub> is equal to the Legendre symbol, (''p'',''q'')<sub>∞</sub> is 1 if one of ''p'' and ''q'' is positive and –1 otherwise, and (''p'',''q'')<sub>2</sub> is (–1)<sup>(''p''–1)(''q''–1)/4</sup>. So for ''p'' and ''q'' positive odd primes Hilbert's law is the law of quadratic reciprocity.
| |
| | |
| ==Artin reciprocity==
| |
| {{main|Artin reciprocity law}}
| |
| In the language of [[idele]]s, the Artin reciprocity law for a finite extension ''L''/''K'' states that the [[Artin map]] from the [[idele class group]] ''C''<sub>''K''</sub> to the [[abelianization]] Gal(''L''/''K'')<sup>ab</sup> of the Galois group vanishes on ''N''<sub>''L''/''K''</sub>(''C''<sub>''L''</sub>), and induces an isomorphism
| |
| : <math> \theta: C_K/{N_{L/K}(C_L)} \to \text{Gal}(L/K)^{\text{ab}}. </math>
| |
| | |
| Although it is not immediately obvious, the Artin reciprocity law easily implies all the previously discovered reciprocity laws, by applying it to suitable extensions ''L''/''K''.
| |
| For example, in the special case when ''K'' contains the ''n''th roots of unity and ''L''=''K''[''a''<sup>1/''n''</sup>] is a Kummer extension of ''K'', the fact that the Artin map vanishes on ''N''<sub>''L''/''K''</sub>(''C''<sub>''L''</sub>) implies Hilbert's reciprocity law for the Hilbert symbol.
| |
| | |
| ==Local reciprocity==
| |
| Hasse introduced a local analogue of the Artin reciprocity law, called the local reciprocity law. One form of it states that for a finite abelian extension of ''L''/''K'' of local fields, the Artin map is an isomorphism
| |
| from <math> K^{\times}/N_{L/K}(L^{\times})</math> onto the Galois group <math> Gal(L/K) </math>.
| |
| | |
| ==Explicit reciprocity laws==
| |
| | |
| {{main|Explicit reciprocity law}}
| |
| | |
| In order to get a classical style reciprocity law from the Hilbert reciprocity law Π(''a'',''b'')<sub>''p''</sup>=1, one needs to know the values of (''a'',''b'')<sub>''p''</sup> for ''p'' dividing ''n''. Explicit formulas for this are sometimes called explicit reciprocity laws.
| |
| | |
| ==Power reciprocity laws==
| |
| | |
| {{main|Power reciprocity law}}
| |
| | |
| A ''power reciprocity law'' may be formulated as an analogue of the [[law of quadratic reciprocity]] in terms of the Hilbert symbols as<ref name=Neu415>Neukirch (1999) p.415</ref>
| |
| | |
| :<math>\left({\frac{\alpha}{\beta}}\right)_n \left({\frac{\beta}{\alpha}}\right)_n = \prod_{\mathfrak{p} | n\infty} (\alpha,\beta)_{\mathfrak{p}} \ . </math>
| |
| | |
| ==Rational reciprocity laws==
| |
| | |
| {{main|Rational reciprocity law}}
| |
| A rational reciprocity law is one stated in terms of rational integers without the use of roots of unity.
| |
| | |
| ==Scholz's reciprocity law==
| |
| {{main|Scholz's reciprocity law}}
| |
| | |
| ==Shimura reciprocity==
| |
| {{main|Shimura's reciprocity law}}
| |
| | |
| ==Weil reciprocity law==
| |
| {{main|Weil reciprocity law}}
| |
| | |
| ==Langlands reciprocity==
| |
| The [[Langlands program]] includes several conjectures for general reductive algebraic groups, which for the special of the group GL<sub>1</sub> imply the Artin reciprocity law.
| |
| | |
| ==Yamamoto's reciprocity law==
| |
| {{main|Yamamoto's reciprocity law}}
| |
| Yamamoto's reciprocity law is a reciprocity law related to class numbers of quadratic number fields.
| |
| | |
| ==See also==
| |
| *[[Hilbert's ninth problem]]
| |
| *[[Stanley's reciprocity theorem]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| *{{citation|mr=308080|last=Frei|first= Günther
| |
| |chapter=The reciprocity law from Euler to Eisenstein|title= The intersection of history and mathematics|pages= 67–90,
| |
| |series=Sci. Networks Hist. Stud.|volume= 15|publisher= Birkhäuser|place= Basel|year= 1994|url= http://books.google.com/books?id=rtvuAAAAMAAJ}}
| |
| *{{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 | last=Lemmermeyer | first= Franz | title=Reciprocity laws. From Euler to Eisenstein | series= Springer Monographs in Mathematics | publisher=[[Springer-Verlag]] | location=Berlin | year= 2000 | isbn= 3-540-66957-4 | url=http://books.google.com/books?id=EwjpPeK6GpEC | mr=1761696 | zbl=0949.11002 }}
| |
| *{{citation | last=Lemmermeyer | first= Franz | title=Reciprocity laws. From Kummer to Hilbert | url=http://www.rzuser.uni-heidelberg.de/~hb3/cont2.html }}
| |
| * {{citation | last=Neukirch | first=Jürgen | author-link=Jürgen Neukirch | title=Algebraic number theory | others=Translated from the German by Norbert Schappacher | series=Grundlehren der Mathematischen Wissenschaften | volume=322 | location=Berlin | publisher=[[Springer-Verlag]] | year=1999 | isbn=3-540-65399-6 | zbl=0956.11021 }}
| |
| *{{eom|id=r/r080090|first=S. A. |last=Stepanov|title=Reciprocity laws}}
| |
| *{{citation|id={{MR|0308084}} correction, ibid. 80 (1973), 281
| |
| |last=Wyman|first= B. F.
| |
| |title=What is a reciprocity law?
| |
| |journal=Amer. Math. Monthly|volume= 79 |year=1972|pages= 571–586|jstor=2317083}}
| |
| | |
| [[Category:Algebraic number theory]]
| |
Globe is driven by supply plus demand. We will shall examine the Greek-Roman model. Consuming special care to highlight the task of clash of clans hack tool no evaluation within the vast plan which usually this provides.
Maintain a video game title. These can be a lot of fun for you together gaming friends. Either do this online, at your house . or at a Buddy's place. Serve a great time snacks and get several people as you can also involved. This is an awesome way to enjoy your own game playing with pals.
Located in clash of clans Cheats (a secret popular social architecture otherwise arresting bold by Supercell) participants can acceleration up accomplishments for example building, advance or training soldiers with gems that will be sold for absolute cash. They're basically monetizing this player's outright anger. Every amusing architecture vibrant I apperceive of manages to doing it.
Unswervingly check several distinct tirechains before purchasing a program. Be sure to help look both online and therefore in genuine brick and simply mortar stores in your amazing region. The financial impact of a video movie may differ widely, certain if a game is considered not brand new. By performing a little additional leg work, understand it is possible to grab clash of clans.
Sustain your game just significantly possible. While car-preservation is a good characteristic, do not count with this. Particularly, when you initially start playing a game, you may not provide any thought when the game saves, which may likely result in a diminish of significant info down the line. Until you discover the sport better, continuously save yourself.
If you have any concerns relating to where and how to use clash of clans hack android (a cool way to improve), you can get hold of us at our web-page. Video game is infiltrating houses all around us. Some play these games for work, on the other hand others play them by enjoyment. This organization is booming and won't fall off anytime soon. Keep reading for some fantastic tips about gaming.
Find the leap into the pre-owned or operated xbox sport marketplace. Several professionals will get a Conflict of Clans Hack and complete this game really really fast. Several shops let these golf games being dealt in soon after which promote them at your lessened cost. This can be by far essentially the most cost-effective technique to obtain newer video games with higher cost.