The simplest manifestation of reciprocity laws is the following fact, which was already known to P. Fermat. The only prime divisors of the numbers are and primes which are terms of the arithmetical series . In other words, the identity
where is a prime, is solvable if and only if . This assertion may be expressed with the aid of the quadratic-residue symbol (Legendre symbol) as follows:
In the more general case, the problem of solvability of the congruence
is solved by the Gauss reciprocity law:
where and are different odd primes, and by the following two complements:
These relations for the Legendre symbol show that the prime numbers for which (*) is solvable for a given non-square are contained in exactly one-half of the residue classes modulo .
C.F. Gauss recognized the great importance of this reciprocity law and gave several proofs of it, based on completely different concepts . It follows from Gauss' reciprocity law and from its further generalization (the reciprocity law for the Jacobi symbol) that, in particular, the decomposition of a prime number in a quadratic extension of the field of rational numbers (cf. Quadratic field) is determined by the residue class of modulo .
Gauss' reciprocity law has been generalized to congruences of the form
However, this involves a transition from the arithmetic of the rational integers to the arithmetic of the integers of an extension of finite degree of the field of rational numbers. Also, in generalizing the reciprocity law to -th power residues, the extension must be assumed to contain a primitive -th root of unity . Under this assumption, prime divisors of which are not divisors of satisfy the congruence
where is the norm of the divisor , equal to the number of residue classes of the maximal order of this field modulo . The analogue of the Legendre symbol is defined by the congruence
The power-residue symbol for a pair of integers and , analogous to the Jacobi symbol, is defined by the formula
if is the decomposition of the principal divisor into prime factors and and are relatively prime.
The reciprocity law for in the field was established by Gauss , while that for in the field was established by G. Eisenstein . E. Kummer  established the general reciprocity law for the power-residue symbol in the field , where is a prime. Kummer's formula for a regular prime number has the form
where are integers in the field ,
and is a polynomial of degree such that
The next stage in the study of general reciprocity laws is represented by the work of D. Hilbert , , 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:
He also noted the analogy between this formula and the theorem on residues of algebraic functions — regular points with norm-residue symbol correspond to branch points on a Riemann surface.
Further advances in the study of reciprocity laws are due to Ph. Furtwängler , T. Takagi , E. Artin , and H. Hasse . The most general form of the reciprocity law was obtained by I.R. Shafarevich .
Similarly to Gauss' reciprocity law, the general reciprocity law is closely connected with the study of decomposition laws of prime divisors of a given algebraic number field in an algebraic extension with an Abelian Galois group. In particular, class field theory, which offers a solution to this problem, may be based  on Shafarevich's reciprocity law.
|||C.F. Gauss, "Untersuchungen über höhere Arithmetik" , Springer (1889) (Translated from Latin)|
|||C.F. Gauss, "Theoria residuorum biquadraticorum" , Werke , 2 , K. Gesellschaft Wissenschaft. Göttingen (1876) pp. 65|
|||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|
|||E.E. Kummer, "Allgemeine Reciprocitätsgesetze für beliebig hohe Potentzreste" Ber. K. Akad. Wiss. Berlin (1850) pp. 154–165|
|||D. Hilbert, "Die Theorie der algebraischen Zahlkörper" Jahresber. Deutsch. Math.-Verein , 4 (1897) pp. 175–546|
|||D. Hilbert, "Ueber die theorie der relativquadratischen Zahlkörpern" Jahresber. Deutsch. Math.-Verein , 6 : 1 (1899) pp. 88–94|
|[7a]||Ph. Furtwängler, "Die Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern (Erster Teil)" Math. Ann. , 67 (1909) pp. 1–31|
|[7b]||Ph. Furtwängler, "Die Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern (Zweiter Teil)" Math. Ann. , 72 (1912) pp. 346–386|
|[7c]||Ph. Furtwängler, "Die Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern (Dritter und letzter Teil)" Math. Ann. , 74 (1913) pp. 413–429|
|||T. Takagi, "Ueber eine Theorie der relativ Abel'schen Zahlkörpers" J. Coll. Sci. Tokyo , 41 : 9 (1920) pp. 1–133|
|||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))|
|||H. Hasse, "Die Struktur der R. Brauerschen Algebrenklassengruppe über einen algebraischer Zahlkörper" Math. Ann. , 107 (1933) pp. 731–760|
|||I.R. Shafarevich, "A general reciprocity law" Uspekhi Mat. Nauk , 3 : 3 (1948) pp. 165 (In Russian)|
|||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)|
|||J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)|
|||D.K. Faddeev, "On Hilbert's ninth problem" , Hilbert problems , Moscow (1969) pp. 131–140 (In Russian)|
|[a1]||J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. 4, §4|
Reciprocity laws. S.A. Stepanov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Reciprocity_laws&oldid=14911