174 III More on Divisibility
5 FurtherRemarks
For the history of the law of quadratic reciprocity, see Frei [16]. The first two proofs by
Gauss of the law of quadratic reciprocity appeared in§§125–145 and§262 of [17]. A
simplified account of Gauss’s inductive proof has been given by Brown [7]. The proofs
most commonly given use ‘Gauss’s lemma’ and are variants of Gauss’s third proof.
The first proof given here, due to Rousseau [46], is of this general type, but it does not
use Gauss’s lemma and is based on a natural definition of the Jacobi symbol. For an
extension of this definition of Zolotareff to algebraic number fields, see Cartier [9].
For Dirichlet’s evaluation of Gauss sums, see [33]. A survey of Gauss sums is given
in Berndt and Evans [6].
The extension of the law of quadratic reciprocity to arbitrary algebraic number
fields was the subject of Hilbert’s 9th Paris problem. Although such generalizations lie
outside the scope of the present work, it may be worthwhile to give a brief glimpse.
LetK=Qbe the field of rational numbers and letL=Q(
√
d)be a quadratic exten-
sion ofK.Ifpis a prime inK, the law of quadratic reciprocity may be interpreted as
describing how the ideal generated bypinLfactors into prime ideals. Now letKbe
an arbitrary algebraic number field and letLbe any finite extension ofK. Quite gener-
ally, we may ask how the arithmetic of the extensionLis determined by the arithmetic
ofK. The general reciprocity law, conjectured by Artin in 1923 and proved by him
in 1927, gives an answer in the form of an isomorphism between two groups, provided
the Galois group ofLoverKis abelian. For an introduction, see Wyman [54] and, for
more detail, Tate [51]. The outstanding problem is to find a meaningful extension to the
case when the Galois group is non-abelian. Some intriguing conjectures are provided
by the Langlands program, for which see also Gelbart [18].
The law of quadratic reciprocity has an analogue for polynomials with coefficients
from a finite field. LetFqbe a finite field containingqelements, whereqis a power
of anoddprime. Ifg∈Fq[x] is a monic irreducible polynomial of positive degree,
then for anyf∈Fq[x] not divisible bygwe define(f/g)to be 1 iffis congruent to
a square modg,and−1 otherwise. The law of quadratic reciprocity, which in the case
of primeqwas stated by Dedekind (1857) and proved by Artin (1924), says that
(f/g)(g/f)=(− 1 )mn(q−^1 )/^2for any distinct monic irreducible polynomialsf,g∈Fq[x] of positive degreesm,n.
Artin also developed a theory of ideals, analogous to that for quadratic number fields,
for the field obtained by adjoining toFq[x]anelementωwithω^2 = D(x),where
D(x)∈Fq[x] is square-free; see [3].
Quadratic fields are treated in the third volume of Landau [30]. There is also a
useful resum ́e accompanying the tables in Ince [23].
A complex number is said to bealgebraicif it is a root of a monic polynomial
with rational coefficients andtranscendentalotherwise. Hence a complex number is
algebraic if and only if it is an element of some algebraic number field.
For an introduction to the theory of algebraic number fields, see Samuel [47]. This
vast theory may be approached in a variety of ways. For a more detailed treatment
the student may choose from Hecke [22], Hasse [20], Lang [32], Narkiewicz [38] and
Neukirch [39]. There are useful articles in Cassels and Fr ̈ohlich [10], and Artin [2]
treats also algebraic functions.