View source for Reciprocity laws

A number of statements expressing relations between power-residue symbols or norm-residue symbols (cf. [[Power residue|Power residue]]; [[Norm-residue symbol|Norm-residue symbol]]).

The simplest manifestation of reciprocity laws is the following fact, which was already known to P. Fermat. The only prime divisors of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r0800901.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r0800902.png" /> and primes which are terms of the arithmetical series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r0800903.png" />. In other words, the identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r0800904.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r0800905.png" /> is a prime, is solvable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r0800906.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r0800907.png" />. This assertion may be expressed with the aid of the quadratic-residue symbol ([[Legendre symbol|Legendre symbol]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r0800908.png" /> as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r0800909.png" /></td> </tr></table>

In the more general case, the problem of solvability of the congruence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

is solved by the [[Gauss reciprocity law|Gauss reciprocity law]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009013.png" /> are different odd primes, and by the following two complements:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009014.png" /></td> </tr></table>

These relations for the Legendre symbol show that the prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009015.png" /> for which (*) is solvable for a given non-square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009016.png" /> are contained in exactly one-half of the residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009017.png" />.

C.F. Gauss recognized the great importance of this reciprocity law and gave several proofs of it, based on completely different concepts [[#References|[1]]]. It follows from Gauss' reciprocity law and from its further generalization (the reciprocity law for the [[Jacobi symbol|Jacobi symbol]]) that, in particular, the decomposition of a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009018.png" /> in a quadratic extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009019.png" /> of the field of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009020.png" /> (cf. [[Quadratic field|Quadratic field]]) is determined by the residue class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009021.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009022.png" />.

Gauss' reciprocity law has been generalized to congruences of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009023.png" /></td> </tr></table>

However, this involves a transition from the arithmetic of the rational integers to the arithmetic of the integers of an extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009024.png" /> of finite degree of the field of rational numbers. Also, in generalizing the reciprocity law to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009025.png" />-th power residues, the extension must be assumed to contain a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009026.png" />-th root of unity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009027.png" />. Under this assumption, prime divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009029.png" /> which are not divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009030.png" /> satisfy the congruence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009031.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009032.png" /> is the norm of the divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009033.png" />, equal to the number of residue classes of the maximal order of this field modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009034.png" />. The analogue of the Legendre symbol is defined by the congruence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009035.png" /></td> </tr></table>

The power-residue symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009036.png" /> for a pair of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009038.png" />, analogous to the Jacobi symbol, is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009039.png" /></td> </tr></table>

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009040.png" /> is the decomposition of the principal divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009041.png" /> into prime factors and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009043.png" /> are relatively prime.

The reciprocity law for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009044.png" /> in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009045.png" /> was established by Gauss [[#References|[2]]], while that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009046.png" /> in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009047.png" /> was established by G. Eisenstein [[#References|[3]]]. E. Kummer [[#References|[4]]] established the general reciprocity law for the power-residue symbol in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009049.png" /> is a prime. Kummer's formula for a [[Regular prime number|regular prime number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009050.png" /> has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009051.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009052.png" /> are integers in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009053.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009054.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009055.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009056.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009057.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009058.png" /></td> </tr></table>

The next stage in the study of general reciprocity laws is represented by the work of D. Hilbert [[#References|[5]]], [[#References|[6]]], who cleared up their local aspect. He established, in certain cases, reciprocity laws in the form of a product formula for his norm-residue symbol:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009059.png" /></td> </tr></table>

He also noted the analogy between this formula and the theorem on residues of algebraic functions — regular points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009060.png" /> with norm-residue symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009061.png" /> correspond to branch points on a Riemann surface.

Further advances in the study of reciprocity laws are due to Ph. Furtwängler , T. Takagi [[#References|[8]]], E. Artin [[#References|[9]]], and H. Hasse [[#References|[10]]]. The most general form of the reciprocity law was obtained by I.R. Shafarevich [[#References|[11]]].

Similarly to Gauss' reciprocity law, the general reciprocity law is closely connected with the study of decomposition laws of prime divisors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009062.png" /> of a given algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009063.png" /> in an algebraic extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080090/r08009064.png" /> with an Abelian Galois group. In particular, [[Class field theory|class field theory]], which offers a solution to this problem, may be based [[#References|[12]]] on Shafarevich's reciprocity law.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,   "Untersuchungen über höhere Arithmetik" , Springer  (1889)  (Translated from Latin)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.F. Gauss,   "Theoria residuorum biquadraticorum" , ''Werke'' , '''2''' , K. Gesellschaft Wissenschaft. Göttingen  (1876)  pp. 65</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Eisenstein,   "Beweis des Reciprocitätssatzes für die cubischen Reste in der Theorie der aus dritten Würzeln der Einheit zusammengesetzten complexen Zahlen"  ''J. Math.'' , '''27'''  (1844)  pp. 289–310</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.E. Kummer,   "Allgemeine Reciprocitätsgesetze für beliebig hohe Potentzreste"  ''Ber. K. Akad. Wiss. Berlin''  (1850)  pp. 154–165</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D. Hilbert,   "Die Theorie der algebraischen Zahlkörper"  ''Jahresber. Deutsch. Math.-Verein'' , '''4'''  (1897)  pp. 175–546</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D. Hilbert,   "Ueber die theorie der relativquadratischen Zahlkörpern"  ''Jahresber. Deutsch. Math.-Verein'' , '''6''' :  1  (1899)  pp. 88–94</TD></TR><TR><TD valign="top">[7a]</TD> <TD valign="top">  Ph. Furtwängler,   "Die Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern (Erster Teil)"  ''Math. Ann.'' , '''67'''  (1909)  pp. 1–31</TD></TR><TR><TD valign="top">[7b]</TD> <TD valign="top">  Ph. Furtwängler,   "Die Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern (Zweiter Teil)"  ''Math. Ann.'' , '''72'''  (1912)  pp. 346–386</TD></TR><TR><TD valign="top">[7c]</TD> <TD valign="top">  Ph. Furtwängler,   "Die Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern (Dritter und letzter Teil)"  ''Math. Ann.'' , '''74'''  (1913)  pp. 413–429</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  T. Takagi,   "Ueber eine Theorie der relativ Abel'schen Zahlkörpers"  ''J. Coll. Sci. Tokyo'' , '''41''' :  9  (1920)  pp. 1–133</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  E. Artin,   "Beweis des allgemeinen Reziprocitätsgesetzes"  ''Abh. Math. Sem. Univ. Hamburg'' , '''5'''  (1928)  pp. 353–363  ((also: Collected Papers, Addison-Wesley, 1965, pp. 131–141))</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  H. Hasse,   "Die Struktur der R. Brauerschen Algebrenklassengruppe über einen algebraischer Zahlkörper"  ''Math. Ann.'' , '''107'''  (1933)  pp. 731–760</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  I.R. Shafarevich,   "A general reciprocity law"  ''Uspekhi Mat. Nauk'' , '''3''' :  3  (1948)  pp. 165  (In Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  A.I. Lapin,   "A general law of dependence and a new foundation of class field theory"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''18'''  (1954)  pp. 335–378  (In Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  D.K. Faddeev,   "On Hilbert's ninth problem" , ''Hilbert problems'' , Moscow  (1969)  pp. 131–140  (In Russian)</TD></TR></table>



====Comments====
For a discussion of reciprocity laws in the context of modern class field theory see [[#References|[a1]]] and [[Class field theory|Class field theory]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Neukirch,   "Class field theory" , Springer  (1986)  pp. Chapt. 4, §4</TD></TR></table>