Reciprocity laws

A number of statements expressing relations between power-residue symbols or norm-residue symbols (cf. Power residue; 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) $\left ( \frac{a}{p} \right )$ 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:

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

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 [1]. 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 $p$ in a quadratic extension $\mathbf Q ( \sqrt d )$ of the field of rational numbers $\mathbf Q$( cf. 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 [2], while that for $n = 3$ in the field $\mathbf Q ( e ^ {2 \pi i / 3 } )$ was established by G. Eisenstein [3]. E. Kummer [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 $n$ 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 [5], [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 [8], E. Artin [9], and H. Hasse [10]. The most general form of the reciprocity law was obtained by I.R. Shafarevich [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, which offers a solution to this problem, may be based [12] on Shafarevich's reciprocity law.

References

[1]	C.F. Gauss, "Untersuchungen über höhere Arithmetik" , Springer (1889) (Translated from Latin)
[2]	C.F. Gauss, "Theoria residuorum biquadraticorum" , Werke , 2 , K. Gesellschaft Wissenschaft. Göttingen (1876) pp. 65
[3]	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
[4]	E.E. Kummer, "Allgemeine Reciprocitätsgesetze für beliebig hohe Potentzreste" Ber. K. Akad. Wiss. Berlin (1850) pp. 154–165
[5]	D. Hilbert, "Die Theorie der algebraischen Zahlkörper" Jahresber. Deutsch. Math.-Verein , 4 (1897) pp. 175–546
[6]	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
[8]	T. Takagi, "Ueber eine Theorie der relativ Abel'schen Zahlkörpers" J. Coll. Sci. Tokyo , 41 : 9 (1920) pp. 1–133
[9]	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))
[10]	H. Hasse, "Die Struktur der R. Brauerschen Algebrenklassengruppe über einen algebraischer Zahlkörper" Math. Ann. , 107 (1933) pp. 731–760
[11]	I.R. Shafarevich, "A general reciprocity law" Uspekhi Mat. Nauk , 3 : 3 (1948) pp. 165 (In Russian)
[12]	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)
[13]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)
[14]	D.K. Faddeev, "On Hilbert's ninth problem" , Hilbert problems , Moscow (1969) pp. 131–140 (In Russian)

Comments

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

References

[a1]	J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. 4, §4