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In mathematics, an '''explicit reciprocity law''' is a formula for the [[Hilbert symbol]] of a [[local field]]. The name "explicit reciprocity law" refers to the fact that the Hilbert symbols of local fields appear in [[Hilbert's reciprocity law]] for the [[power residue symbol]]. The definitions of the Hilbert symbol are usually rather roundabout and can be hard to use directly in explicit examples, and the explicit reciprocity laws give more explicit expressions for the Hilbert symbol that are sometimes easier to use.

There are also several explicit reciprocity laws for various generalizations of the Hilbert symbol to [[higher local field]]s, [[p-divisible group|''p''-divisible group]]s, and so on.

==History==
{{harvtxt|Artin|Hasse|1928}} gave an explicit formula for the Hilbert symbol (α,β) in the case of odd prime powers, for some special values of α and β  when the field is the (cyclotomic) extension of the ''p''-adic numbers by a ''p''<sup>''n''</sup>th root of unity. {{harvtxt|Iwasawa|1968}} extended the formula of Artin and Hasse to more cases of α and β, and {{harvtxt|Wiles|1978}} and {{harvtxt|de Shalit|1986}} extended Iwasawa's work to [[Lubin–Tate extension]]s of local fields. 
{{harvtxt|Shafarevich|1950}} gave an explicit formula for the Hilbert symbol for odd prime powers for general local fields. His formula was rather complicated which made it hard to use, and {{harvs|txt|last=Brückner|year1=1967|year2=1979}} and {{harvtxt|Vostokov|1978}} found a simpler formula. {{harvtxt|Henniart|1981}} simplified Vostokov's work and extended it to the case of even prime powers.

==Examples==

For archimedean local fields or in the unramified case the Hilbert symbol is easy to write down explicitly. The main problem is to evaluate it in the ramified case. 

===Archimedean fields===

Over the complex numbers (''a'', ''b'') is always 1.
Over the reals, the Hilbert symbol of odd degree is trivial, and the Hilbert symbol of even degree is given by  (''a'', ''b'')<sub>&infin;</sub> is +1 if at least one of ''a'' or ''b'' is positive, and &minus;1 if both are negative.

===Unramified case: the tame Hilbert symbol===
In the unramified case, when the order of the Hilbert symbol is coprime to the residue characteristic of the local field, the ''tame Hilbert symbol'' is given by<ref name=Neu335>Neukirch (1999) p.335</ref>

:<math> (a,b) = \omega((-1)^{\operatorname{ord}(a)\operatorname{ord}(b)}b^{\operatorname{ord}(a)}/a^{\operatorname{ord}(b)})^{(q-1)/n}</math>

where ω(''a'') is the (''q''&nbsp;–&nbsp;1)-th root of unity congruent to ''a'' and ord(''a'') is the value of the valuation of the local field, and ''n'' is the degree of the Hilbert symbol, and ''q'' is the order of the residue class field. The number ''n'' divides ''q''&nbsp;–&nbsp;1 because the local field contains the ''n''th roots of unity by assumption. 

As a special case, over the p-adics with ''p'' odd, writing <math>a = p^\alpha u</math> and <math>b = p^\beta v</math>, where ''u'' and ''v'' are integers coprime to ''p'', we have for the quadratic Hilbert symbol

:<math>(a,b)_p = (-1)^{\alpha\beta\varepsilon(p)} \left(\frac{u}{p}\right)^\beta \left(\frac{v}{p}\right)^\alpha</math>, where <math>\varepsilon(p) = (p-1)/2</math> 

and the expression involves two [[Legendre symbol]]s.

===Ramified case===

The simplest example of a Hilbert symbol in the ramified case is the quadratic Hilbert symbol over the 2-adic integers.
Over the 2-adics, again writing <math>a = 2^\alpha u</math> and <math>b = 2^\beta v</math>, where ''u'' and ''v'' are [[odd number]]s, we have for the quadratic Hilbert symbol

:<math>(a,b)_2 = (-1)^{\epsilon(u)\epsilon(v) + \alpha\omega(v) + \beta\omega(u)}</math>, where <math>\omega(x) = (x^2-1)/8.</math>

==See also==
* [[Rational reciprocity law]]

==Notes==
{{reflist}}

==References==

*{{citation|jfm= 54.0191.05 |last1=Artin|first1= E.|last2= Hasse|first2= H.
|title=Die beiden Ergänzungssätze zum Reziprozitätsgesetz der ''l''<sup>''n''</sup>-ten Potenzreste im Körper der ''l''<sup>''n''</sup>-ten Einheitswurzeln|journal=Abhandlungen Hamburg |volume=6|pages= 146–162 |year=1928}}
*{{citation|id={{MR|0230702}}
|last=Brückner|first= Helmut
|chapter=Eine explizite Formel zum Reziprozitätsgesetz für Primzahlexponenten p|language=German|year= 1967 |title=Algebraische Zahlentheorie (Ber. Tagung Math. Forschungsinst. Oberwolfach, 1964) |pages= 31–39 |publisher=Bibliographisches Institut, Mannheim}}
*{{citation|id={{MR|0533354}}|last=Brückner|first= H.|title=Explizites Reziprozitätsgesetz und Anwendungen|language=German|series=
Vorlesungen aus dem Fachbereich Mathematik der Universität Essen |volume= 2| publisher= Universität Essen, Fachbereich Mathematik, Essen, |year=1979}}
*{{citation|id={{MR|0835803}}|last=de Shalit|first= Ehud|title=The explicit reciprocity law in local class field theory|journal=Duke Math. J. |volume=53 |year=1986|issue= 1|pages= 163–176|url=http://projecteuclid.org/euclid.dmj/1077304887}}
*{{citation|id={{MR|0636453}}|last=Henniart|first= Guy
|title=Sur les lois de réciprocité explicites. I. |language=French
|journal=J. Reine Angew. Math.|volume= 329 |year=1981|pages= 177–203|url=http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PPN=PPN243919689_0329}} 
*{{citation|id={{MR|0229609}}|last=Iwasawa|first= Kenkichi|title=On explicit formulas for the norm residue symbol|journal= J. Math. Soc. Japan |volume=20 |year=1968|pages= 151–165}} 
* {{Neukirch ANT}}
*{{citation|id={{MR|0031944}}|last=Shafarevich|first= I. R.|title=A general reciprocity law|language=Russian|journal=
Mat. Sbornik N.S.|volume= 26|year=1950|pages= 113–146}} 
*{{citation|id={{MR|0522940}}|last=Vostokov|first= S. V.|title=An explicit form of the reciprocity law|journal=Izv. Akad. Nauk SSSR Ser. Mat.|volume= 42 |year=1978|issue= 6|pages= 1288–1321, 1439|doi=10.1070/IM1979v013n03ABEH002077}} 
*{{citation|id={{MR|0480442}}|last=Wiles|first= A.|title=Higher explicit reciprocity laws|journal=
Annals of Mathematics |volume= 107 |year=1978|issue= 2|pages=235–254|url= http://www.jstor.org/stable/1971143}}

==Further reading==
* {{citation | last=Lemmermeyer | first=Franz | title=Reciprocity laws. From Euler to Eisenstein | series=Springer Monographs in Mathematics | location=Berlin | publisher=[[Springer-Verlag]] | year=2000 | isbn=3-540-66957-4 | zbl=0949.11002 | mr=1761696 }}

[[Category:Algebraic number theory]]