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.

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

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