Reciprocity Laws

In this first chapter we will present the fathers of the quadratic reciprocity law. Although some results on quadratic residues modulo 10 have been found very early on (see [Ene]) — in connection with the problem of characterizing perfect squares — the history of modern number theory starts with the editions of the books of Diophantus, in particular with the commented edition by Bachet in 1621.

The aim of this chapter is to present those proofs of the quadratic reciprocity law which are based on the theory of quadratic number fields. The first proof using such techniques was Gauss’s second proof; instead of developing the theory of binary quadratic forms we will give a proof using the ideal theoretic language. After developing the complete genus theory for quadratic number fields, we give some applications to the primality tests of Lucas-Lehmer.

This chapter is devoted to some proofs of the quadratic reciprocity law that make use of the arithmetic of cyclotomic number fields.

Rational reciprocity deals with residue symbols which assume only the values ±1 and which have entries in ℤ; the first rational reciprocity laws other than quadratic reciprocity were discovered by Dirichlet. His results, however, were soon forgotten and have been rediscovered regularly.

In Chapter 5 we have already seen a lot about quartic reciprocity and its applications to rational number theory; these rational laws, however, do not suffice to solve every “rational” problem where quartic reciprocity is involved, as the following example shows.

The cubic reciprocity law can be studied in the same way as quartic reciprocity; both laws live in imaginary quadratic number fields with a finite number of units. We will proceed exactly as in the quartic case: first we study connections with the splitting of primes in certain cubic cyclic extensions, and then we use cubic Gauss and Jacobi sums to derive the cubic reciprocity law (expressed less euphemistically: this chapter won’t contain any new ideas — we just compute).

In this chapter we will have a closer look at Eisenstein’s analytic proofs for the reciprocity laws for quadratic, cubic and quartic residues. In the historical survey which he put at the beginning of his paper [468], Kummer praises them with the following words:

Für einen der schönsten Beweise dieses von den ausgezeichnetsten Mathematikern viel bewiesenen Theorems wird aber derjenige mit Recht gehalten, welchen Eisenstein in Crelle’s Journal, Bd. 29, pag. 177, gegeben hat. In diesem wird das Legendresche Zeichen (p / q) durch Kreisfunktionen so ausgedrückt, daß bei der Vertauschung von p und q dieser Ausdruck, bis auf eine leicht zu bestimmende Änderung im Vorzeichen, ungeändert bleibt. [...] Wenn dieser Eisensteinsche Beweis schon wegen seiner vorzüglichen Eleganz beachtenswerth ist, so wird der Werth desselben noch dadurch erhöht, daß er, wie Eisenstein selbst gezeigt hat, ohne besondere Schwierigkeit auch auf die bi-quadratischen und kubischen Reciprocitätsgesetze angewendet werden kann, wenn anstatt der Kreisfunktionen elliptische Funktionen mit bestimmten Moduln angewendet werden.¹

The proof of the octic reciprocity law differs considerably from those in the cubic and quartic cases; in fact one needs methods that are much more sophisticated than Gauss sums, and this is why we resurrect elliptic Gauss sums These were introduced by Eisenstein while he was working on octic reciprocity and the division of the lemniscate; apparently, nobody ever bothered to study these sums that I wouldn’t have hesitated to call Eisenstein sums were it not for the fact that this name is already being used for the sums that we have studied in Chapter 4. In contrast to the full octic reciprocity law, rational octic reciprocity laws can be proved quite easily: the octic version of Burde’s reciprocity law is presented in Section 9.1, Eisenstein’s octic reciprocity law and the formulas of Western are discussed in Section 9.2, and the proof of Scholz’s octic reciprocity law is given in Section 9.5.

It is well known that Gauss recorded many of his discoveries in a diary; it ends with the ‘Last Entry’ from July 9, 1814, which reads as follows¹ (see [Kle]):

Observatio per inductionem facta gravissima theoriam residuorum biquadraticorum cum functionibus lemniscaticis elegantissime nectens. Puta si a + bi est numerus primus, a − 1 + bi per 2 + 2i divisibilis, multitudo omnium solutionum congruentiae

$$1 \equiv xx + yy + xxyy(\bmod {\text{ }}a + bi) $$

inclusis x = ∞, y = ±i; x = ±i, y = ∞ fit = (a − 1)² + b ².

In order to prove higher reciprocity laws, the methods known to Gauss were soon found to be inadequate. The most obvious obstacle, namely the fact that the unique factorization theorem fails to hold for the rings ℤ[ζ_ℓ], was overcome by Kummer through the invention of his ideal numbers. The direct generalization of the proofs for cubic and quartic reciprocity, however, did not yield the general reciprocity theorem for ℓ-th powers: indeed, the most general reciprocity law that could be proved within the cyclotomic framework is Eisenstein’s reciprocity law. The key to its proof is the prime ideal factorization of Gauss sums; since we can express Gauss sums in terms of Jacobi sums and vice versa, the prime ideal factorization of Jacobi sums would do equally well.