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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  $  x  ^ {2} + 1 $
are  $  2 $
and primes which are terms of the arithmetical series  $  1 + 4k $. 
In other words, the identity

$$ 
x  ^ {2} + 1  \equiv  0  (  \mathop{\rm mod}  p) ,
$$

where  $  p > 2 $
is a prime, is solvable if and only if  $  p \equiv 1 $
$  (  \mathop{\rm mod}  4) $. 
This assertion may be expressed with the aid of the quadratic-residue symbol ([[Legendre symbol|Legendre symbol]])  $  \left ( 
\frac{a}{p}
 \right ) $
as follows:

$$ 
\left ( - 
\frac{1}{p}
 \right )  =  (- 1) ^ {( p- 1) / {2 } } .
$$

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

$$ \tag{* }
x  ^ {2}  \equiv  a  (  \mathop{\rm mod}  p)
$$

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

$$ 
\left ( {
\frac{p}{q}
 } \right ) \left ( {
\frac{q}{p}
 } \right )  = \ 
(- 1) ^ {( p- 1) / 2 \cdot ( q- 1) / 2 } ,
$$

where  $  p $
and  $  q $
are different odd primes, and by the following two complements:

$$ 
\left ( {- 
\frac{1}{p}
 } \right )  = \ 
(- 1) ^ {( p- 2) / {2 } } \ \ 
\textrm{ and } \  \left ( {
\frac{2}{p}
 } \right )  = \ 
(- 1) ^ {( p  ^ {2} - 1) / {8 } } .
$$

These relations for the Legendre symbol show that the prime numbers  $  p $
for which (*) is solvable for a given non-square  $  a $
are contained in exactly one-half of the residue classes modulo  $  4  | a | $.

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  $  p $
in a quadratic extension  $  \mathbf Q ( \sqrt d ) $
of the field of rational numbers  $  \mathbf Q $(
cf. [[Quadratic field|Quadratic field]]) is determined by the residue class of  $  p $
modulo  $  4  | d | $.

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

$$ 
x  ^ {n}  \equiv  a  (  \mathop{\rm mod}  p),\ \ 
n > 2.
$$

However, this involves a transition from the arithmetic of the rational integers to the arithmetic of the integers of an extension  $  K $
of finite degree of the field of rational numbers. Also, in generalizing the reciprocity law to  $  n $-
th power residues, the extension must be assumed to contain a primitive  $  n $-
th root of unity  $  \zeta $. 
Under this assumption, prime divisors  $  \mathfrak P $
of  $  K $
which are not divisors of  $  n $
satisfy the congruence

$$ 
N _ {\mathfrak P}  \equiv  1  (  \mathop{\rm mod}  n),
$$

where  $  N _ {\mathfrak P} $
is the norm of the divisor  $  \mathfrak P $, 
equal to the number of residue classes of the maximal order of this field modulo  $  \mathfrak P $. 
The analogue of the Legendre symbol is defined by the congruence

$$ 
\left ( {
\frac{a}{\mathfrak P}
 } \right )  = \ 
\zeta  ^ {k}  \equiv \ 
a ^ {( N _ {\mathfrak P} - 1) / {n } } \ 
(  \mathop{\rm mod}  \mathfrak P ).
$$

The power-residue symbol  $  \left ( 
\frac{a}{b}
 \right ) $
for a pair of integers  $  a $
and  $  b $, 
analogous to the Jacobi symbol, is defined by the formula

$$ 
\left ( {
\frac{a}{b}
 } \right )  = \ 
\prod \left ( {
\frac{a}{\mathfrak P _ {i} }
 } \right )  ^ {m} _ {i} ,
$$

if  $  ( b) = \prod \mathfrak P _ {i} ^ {m _ {i} } $
is the decomposition of the principal divisor  $  ( b) $
into prime factors and  $  b $
and  $  an $
are relatively prime.

The reciprocity law for  $  n = 4 $
in the field  $  \mathbf Q ( i) $
was established by Gauss [[#References|[2]]], while that for  $  n = 3 $
in the field  $  \mathbf Q ( e ^ {2 \pi i / 3 } ) $
was established by G. Eisenstein [[#References|[3]]]. E. Kummer [[#References|[4]]] established the general reciprocity law for the power-residue symbol in the field  $  \mathbf Q ( e ^ {2 \pi i / n } ) $, 
where  $  n $
is a prime. Kummer's formula for a [[Regular prime number|regular prime number]]  $  n $
has the form

$$ 
\left ( {
\frac{a}{b}
 } \right ) \left ( {
\frac{b}{a}
 } \right )  ^ {-} 1  = \ 
\zeta ^ {l  ^ {1} ( a) l ^ {n- 1 }
( b) - l  ^ {2} ( a) l ^ {n- 2 } ( b) + \dots - l ^ {n- 1 } ( a) l  ^ {1} ( b) } ,
$$

where  $  a, b $
are integers in the field  $  \mathbf Q ( e ^ {2 \pi i / n } ) $,

$$ 
a  \equiv  b  \equiv  1
 (  \mathop{\rm mod}  ( \zeta - 1)),
$$

$$ 
l  ^ {i} ( a)  =  \left [ 
\frac{d  ^ {i}  \mathop{\rm log}  f( e  ^ {u} ) }{du  ^ {i} }
 \right ] _ {u=} 0 ,
$$

and  $  f( t) $
is a polynomial of degree  $  n - 1 $
such that

$$ 
a  =  f ( \zeta ),\  f ( 1)  =  1.
$$

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:

$$ 
\prod _ { \mathfrak P } \left ( 
\frac{a, b }{\mathfrak P }
 \right )  =  1.
$$

He also noted the analogy between this formula and the theorem on residues of algebraic functions — regular points  $  \mathfrak P $
with norm-residue symbol  $  \neq 1 $
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  $  \mathfrak P $
of a given algebraic number field  $  k $
in an algebraic extension  $  K/k $
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>