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The Reciprocity Law from Euler to Eisenstein

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The Reciprocity Law from Euler to Eisenstein The Reciprocity Law from Euler to Eisenstein Gunther Frei o Introduction The Reciprocity Law plays a very central role in number theory. It grew out of the theory of quadratic forms. The Quadratic Reciprocity Law was first formulated by Euler and Legendre and proved by Gauss and partly by Legendre. The search for higher reciprocity laws gave rise to the introduction and study of the Gaussian integers and more generally of algebraic numbers. Analytic methods introduced by Euler and Dirichlet in connection with the study of sets of prime numbers and primes in arithmetic progressions and their generalization by Dedekind and Weber to algebraic number fields led to a general form of the reciprocity law found and proved by Artin. This Reciprocity Law of Artin which can be considered as being an abelian reciprocity law plays a central role in class field theory. It is the starting point for a search of a more general non-abelian reciprocity law, a hint to whose existence is given by Langlands' program. In this paper we will indicate in the First Part how the Quadratic Reciprocity Law developed out from the theory of quadratic forms and how it was generalized to higher reciprocity laws by Euler, Gauss, Jacobi and Eisenstein. In a Second Part we will take up the development of the analytic methods leading from Euler and Dirichlet to Dedekind, Weber and Artin. 1 Quadratic Reciprocity Law 1.1 Diophantos of Alexandria (ca. 250 A. D. ) In his Arithmetica (comprising 13 books), Diophantos implicitly applies the fol­ lowing theorem (see e.g. [Dio-1893]' Problem XIV in Book 6), which contains a part of the First Complementary Law of the Quadratic Reciprocity. Theorem 1.1 (l) If a is an integer of the form a = 4n + 3, then a = x 2 + y2 has no solutions with integers x, y E Z. In particular: 67 C. Sasaki et al. (eds.), The Intersection of History and Mathematics © Birkhäuser Verlag 1994 68 Gunther Frei (2) If p is a prime number of the form p = 4n + 3, then p = x 2 + y2 has no integral solutions x, y E Z. This is equivalent to say: (3) If p of- 2 is a prime number and p = x 2 + y2 has integral solutions x, y E Z, then p must be of the form p = 4n + 1. Remarks: (a) In Problem XIV of Book 6 Diophantos applies the Theorem 1.1(1) to a = 15. (b) Theorem 1.1 was explicitly stated for the first time by Fermat in a letter to Mersenne in 1638. (c) Diophantos does not give a proof for this theorem. However, a proof can easily be given by the theory of even and odd numbers developed by the Pythagoreans and which was known to Diophantos. If we use Euler's theorem that Z/pZ is a field if p is a prime number (1775), and Legendre's symbol, introduced by Legendre in 1798 (see Definition 1.16), we can deduce immediately the following form of this theorem. Theorem 1.1 (4) If p of- 2 is a prime number and p = x 2 + y2 has a solution with integers x, y E Z, then ( -;1 ) = 1. Hence we have the following diagram of logical implications. Let p of- 2 be a prime number. p = 4n + 1 (3) / x 2 + y2 = P is solvable with x, y E Z 1): (5) (4)~ The equivalence (5) is the First Complementary Law of the Quadratic Reci­ procity. It was first stated by Fermat in 1640 and first proved by Euler in 1750. 1.2 Fermat (1601-1665) From 1636 on Fermat began studying carefully the new edition of Diophantos' Arithmetica, written in Greek and edited by Bachet (1581-1638) in 1621 together with a Latin translation and a commentary. Motivated by this book Fermat found the following essential part of the First Complementary Law of the Quadratic Reciprocity as stated in a letter to his friend Mersenne (1588-1648) on 25 December 1640. The Reciprocity Law from Euler to Eisenstein 69 Theorem 1.2 If p = 4n + 1 is a prime number, then p = x 2 + y2 has a unique solution with natural numbers x,y E N = {1,2,3, ... }. Similarly, he stated in the same letter: Theorem 1.3 (1) If p = 12n + 1 or p = 12n + 7 is a prime number, then p = x 2 + 3y2 has a unique solution with x, yEN. (2) If p = 8n + 1 or p = 8n + 3 is a prime number, then p = x 2 + 2y2 has a unique solution with x, yEN. Remarks: (a) Fermat does not give any proof for these statements, but he indi­ cates that they can be proved by infinite descent. (b) Proofs for Theorem 1.2 were given by Euler in 1750, 1760 and 1758. Euler also proved a converse of Theorem 1.2 in 1758 which gives a primality criterion for odd numbers m of the form m = 4n + 1. (c) Similarly Euler was able to prove Theorem 1.3(1) in 1763. He also found a converse, stated by Euler in 1759 and 1772, but proved only by Lagrange in 1775. Theorem 1.3(2) was proved by Euler in 1761 and 1774 together with a proof of a converse statement, except for the part that if p = 8n + 3 is a prime, then p = x 2 + 2y2 is uniquely solvable. This part was again first proved by Lagrange in 1775. (d) Let's remark that in modern terminology, a necessary condition for a prime p to be representable as p = x 2 + Ny2 with NEZ and x, yEN is that (-pN) = 1, where C;) is the Legendre symbol (see Definition 1.16). This follows from Euler's theorem that Z/pZ is a field if p is a prime number. (e) We also notice that p = x 2 + Ny2 is solvable with x, yEN for a prime p and a given integer NEZ if and only if p = x 2 + Ny2 = (x + V-Ny)(x - V-Ny) i.e. if and only if p is the norm of an integer ex = x + V-Ny E Z(V-N) = {r + sV-N: r,s E Z} 1.3 Euler (1707-1783) It was Kronecker (1823-1891) who, in 1875, hinted at the fact that the Quadratic Reciprocity Law was already stated by Euler and even stated in essentially the same form as was done later by Legendre (see [Kro-1895]' Vo1.2, 1,1-10). Euler was led to this law by his study of quadratic forms inspired by Fermat's investigations on primes p represented as p = x 2 + Ny2 for N = 1, ±2, 3 with integers x, y E Z. Around 1741, Euler begin his study of the following two problems. 70 Gunther Frei Question 1.4 Given NEZ, describe the primes p # 2 for which p = x 2 + Ny2 is solvable with x, y E Z. A weaker form of this problem is: Question 1.5 Given NEZ, describe the primes p # 2 for which p divides m, plm, where m is any integer of the form m = x 2 + N y2, with x, y E Z. The question 1.4 is more difficult to answer, since it involves the class number of the quadratic field K = Q( V-N). As to the question 1.5, which leads to the reciprocity law, Euler introduced the following notations. Definition 1.6 Let p be a prime, p # 2, and NEZ. plx2 + N y2, i. e. p is a divisor of the form x 2 + N y2, if there exist integers a, b E Z with (a, b) = 1, that is whose greatest common divisor is 1, such that pla2 + Nb2. In order to make it easier for the modern reader, we already introduce the Legendre symbol as defined by Legendre in 1798. Definition 1. 7 Let p # 2 be a prime number and n E Z an integer with (n,p) = 1. (~) := ±1 such that (~) == n(p-l)/2(modulo p). Then we can describe the primes in Question 1.5 in the following way. Proposition 1.8 Let NEZ be given and let p be a prime with (N, p) = 1. Then Proof: This follows from Euler's theorem (1775) that Z/pZ is a field, namely: plx2 + Ny2 ¢? pla2 + Nb2 for some a, b E Z with (a, b) = 1 ¢? a2 + Nb2 == o modulo p ¢? ~~ == -N modulo p ¢? (-::) = 1. Let's introduce for a given integer NEZ the following set of primes. Definition 1.9 PN := {primes p # 2: plx2 + Ny2} = {primes p # 2: (-pN) = I} Question 1.5 then takes on the following form. Question 1.10 Give a description of the set of primes PN for a given NEZ. The Reciprocity Law from Euler to Eisenstein 71 Notice that by the Definition 1.7 the symbol Ap(n) := (~) is a function of n E Z for a given prime p, whereas in the description of PN one has to determine the primes p for which (~) = 1 for a given nEZ, i.e. Xn(P) := (~) appears as a function of p for a given n E Z. The relation between Ap(n) and Xn(P) is the quadratic reciprocity discovered by Euler. Fermat's and Euler's fundamental observation was that the primes p E PN can be described by congruence conditions modulo 41NI. This was first discovered for N = 1,2,3. It already follows from the theorems 1.2 and 1.3, namely: Theorem 1.11 Let p be a prime number, pI- 2.
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