` LA CYCLOTOMIE JADIS ET NAGUERE (CYCLOTOMY PAST AND PRESENT?) ` WEIL BY ANDRE

Literally, ”cyclotomy” signifies ”division of the circle”. The Greek geometers were taught to divide the circle into N equal parts, with ruler and compass, for N of the form 2n , 2n 3, 2n 5, and 2n 15. Euler’s discovery of the relations between trigonometric and exponential functions reduced the problem of the division of the circle to the resolution of equations of the form X n = 1. Gauss, at 19, received the Fields medal (more precisely, he would have received it if it had existed) for having solved the equation X 17 = 1 by a succession of square roots, which implies the possibility of division of the circle into 17 equal parts by ruler and compass. Of course, this result, sensational as it was, was for Gauss only a first step in the theory of binomial equations. 1. It is thus quite rightly that one describes the fields generated over Q by the roots of unity, and their subfields, as ”cyclotomic”, and the word ”cyclotomy” could apply to all that relates to them; one also knows, from Kronecker, that these fields are none other than the abelian extensions of Q. But, since Jacobi and throughout the 19th century, use of this word was reserved (in German, Kreist(h)eilung) for the study of certain remarkable sums of the roots of unity, which nowadays (from Hasse, it seems) are usually called ”Gauss sums”; we will adopt this term, which is convenient, but historically unjustifiable. More precisely, we will agree to name a Gauss sum relative to a finite field Fq of q = pn elements any sum X (1) G = G (χ, ψ) = χ(x)ψ(x) x∈F× q

where χ is a character of the multiplicative group F× q , and ψ is a nontrivial character of the additive group Fq . If ε is a primitve root of X p = 1, the set of values of ψ is { 1, ε, . . . , εp−1 }. If χ is of order m, m divides q − 1, and one can write q − 1 = mv; on will then say that G is of order m; for m = 1, one has G = −1. If r is a generator of the cyclic group F× q , χ is well defined by giving ζ = χ(r), and ζ is a 1

primitive root of Z m = 1; one can then write (2)

G=

q−2 X

i

ζψ r

i




=

m−1 X i=0

i=0

ζ

i

v−1 X


ψ ri+mj .

j=0

The first obvious property of G (χ, ψ) is that it is an algebraic integer in the field Q (ζ, ε), and that all of its conjugates over Q are also Gauss sums; if an automorphism of Q (ζ, ε) maps ζ into ζ t and ε into εu , it maps G (χ, ψ) into G (χt , ψ u ). Moreover, with the evident abuse of notation, one has ψ u (x) = ψ(ux), and consequently: G (χ, ψ u ) = χ (u)−1 G (χ, ψ) ,

(3)

which immediately implies that G (χ, ψ)m is in Q (ζ). We also note right way that one has, for G defined by (1): X X X
GG = (4) ψ (y (z − 1)) χ xy −1 ψ (x − y) = χ (z) x,y

y6=0

z6=0

( X q, if χ 6= 1 χ (z) = =q−1− 1, if χ = 1. z6=0,1 If Fq is the prime field Fp = Z/pZ, one can take ψ(x) = εx , and one will have p−1 p−2 m−1 v=1 X X X X i i+mj (5) G= χ(x)εx = ζ i εr = ζi εr x=1

i=0

i=0

j=0

2.We have gotten ahead of the historical order, to which we presently return. The sums (5) are the particular case of the sums introduced by Lagrange in his great memoir ([1 a]) on the algebraic theory of equations (Galois theory ”before the letter”). It is there where Lagrange shows, among other things, how to generate a cyclic extension of degree m by means of an mth root after adjoining, if necessary, the mth roots of unity (the generator called, quite wrongly, a ”Kummer” generator). He introduces the sums (6)

y = x1 + αx2 + . . . + αm−1 xm ,

where αm = 1, and where x1 , . . . , xm are the roots of an equation of degree m, and he observes that y m is invariant under all cyclic permutations of the xi . He shows for example that one thus ”explains” the classical formulas of resolution by radicals of the equations of the 3rd and 4th degrees. Explaining his method again in his Trait´e of 1808 ([1 b], Note XIII), he gives the sums (6) the name of ”resolvents”, which

remained throughout the 19th century. 3. In 1801, in the 7th section of Disquisitiones ( [2 a] ), Gauss gives a complete description of the ”Galois theory” of Q (ε) considered as a cyclic extension of degree p − 1. He shows in particular that, for p − 1 = mv, Q (ε) posses a sub-field km (unique) of degree m over Q, generated over Q by any of the ”periods of order m”:

(7)

ηi =

v−1 X

εr

i+mj

(0 ≤ i < m),

j=0

these being cyclically permuted by the automorphisms of Q (ε) over Q. The question of the resolution by radicals was too instilled in Gauss’ mind for him to be able to ignore it completely. Either having been informed directly or indirectly of Lagrange’s method (as is probable), or having found it himself (as is possible), he applies it to the intermediate fields between Q and Q (ε); if km is as above, and if k is a subfield of km , it leads to forming the Lagrange resolvents using the ηi and auxiliary roots of unity of order < p. For k = Q, these resolvents are just the sums (5). But Gauss does not seem to attach importance to them; he notes in passing the relation GG = p, and that is just in saying that the extraction of the roots (g m )1/m comes down to a square root and a division of an arc of the circle by m. When a little later Lagrange, in his Trait´e ([1 b], Note XIV) gives an exposition of Gauss’ results based principally on the sums (5), he becomes sharply critical of Gauss, for not having sufficiently accounted for the ambiguity that results from employing the roots of unity of order < p. 4. As Gauss shows, the periods ηi have a multiplication table (8)

η i ηj =

X

Nijk ηk

k

where the Nijk are natural numbers, connected with the numbers of solutions of the congruences AX m + BY m ≡ C (mod p). This fact has important arithmetic consequences, some of which Gauss noticed (for the case m = 3) in the Disq. Later, he developed others for m = 4 ([2 d]). But he was especially interested in the case m = 2, the only case where he believed he could use the ”Gauss sums” in preference to the ”periods” (undoubtedly because then he did not have to introduce

additional irrationality). On then has: (9)

G = η0 − η1 = 1 + 2η0 =

p−1 X

2

εx .

x=0

Here (3) gives G = ±G, thus G2 = ±p from (4), the sign being given by p ≡ ±1 √ (mod 4). It follows that the quadratic field k2 contained in Q (ε) is Q ( ±p). 5. As Gauss indicates in the Disq., this result generalizes to the sum G=

N −1 X

2

αx

x=0

where α is a primitive N th root of unity, with N an arbitrary odd number; one has G2 = ±N , which poses the problem of determining the sign of G, for example for α = e2πi/N ; stated in this form, the problem is not algebraic. ”We observe”, says Gauss in the Disq. (with √ an undoubtedly desired ambiguity) that one always has G = + N √ resp. +i N . In fact, he only obtained the proof of it in 1805; this, published in 1811 ([2 b]) is connected, in a way very obvious to us, to his research (which he did not publish) on the theta functions. For N = pq, with p, q prime, Gauss draws from this his fourth proof of the law of quadratic reciprocity. This work has given rise to, even until very recent times, important generalizations, that we will ignore completely. 6. In 1818, Gauss published his sixth proof of the law of quadratic reciprocity ([2 c]); it is based, among other things, on the Gauss sums of order 2, but considered form a strictly algebraic-arithmetic point of view. Let G be defined by (9). Set q to be an odd prime number 6= p; set p = 2p0 + 1, q = 2q 0 + 1. Using the Legendre symbol, the reciprocity law is written:    −1 0 0 p q · = (−1)p q . q p 0

It was seen that one has G2 = (−1)p p, hence   p p0 q 0 p0 q 0 q−1 q0 G = (−1) · p ≡ (−1) q

(mod q).

But one also has, from the binomial formula: X 2 q  q G ≡ εqx = G (mod q), p x

hence the reciprocity law, since G is relatively prime to q. Of course, Gauss is not allowed (ostensibly) to write the congruences in the ring Z[ε]; he replaces them by congruences modulo (q, 1 + X + . . . + X p−1 ) in the ring Z[X]. Nevertheless, it is incomprehensible that Jacobi, Cauchy, and Eisenstein in turn, published proofs virtually identical to that one (and that they even raised questions of priority between them on this subject) only before Eisentein made the observation that the presentation was practically the same as Gauss’ sixth proof. 7. In a supplement to section VII of the Disq. ([2 e]), Gauss not only gives the proof that GG = p, but also gives the multiplication formula for the Gauss sums. From (1), one can write: X χ(x)χ0 (y)ψ(x + y) G (χ, ψ) G (χ0 , ψ) = x,y6=0

 =

X z6=0



X  X ψ(z)  χ(x)χ0 (−x). χ(x)χ0 (y) + x+y=z x,y6=0

x6=0

Set χ00 = χχ0 . The latter sum is 0 if χ00 6= 1 and (q − 1)χ(−1) if χ00 = 1; the other sum can be written J · G (χ00 , ψ) provided that one sets: X χ(x)χ0 (1 − x). (10) J = J (χ, χ0 ) = x6=0,1

For χ00 = 1, on observes that x 7→ x (1 − x)−1 is a bijection of Fq − { 0, 1 } onto Fq − { 0, −1 }, which gives the value −χ(−1) for J if χ 6= 1 and q − 2 if χ = χ0 = 1, thus, if χ 6= 1:
(11) G (χ, ψ) G χ−1 , ψ = qχ (−1) . The case where χ = 1 or χ0 = 1 is trivial. If χ, χ0 , χ00 are 6= 1, on has (12)

G (χ, ψ) G (χ0 , ψ) = J (χ, χ0 ) · G (χ00 , ψ) .

If ζ is a primitive root of Z m = 1 as above, and if the orders of χ, χ0 are m or divisors of m, (10) shows that J = J (χ, χ0 ) is in Z [ζ]; from (12) and (4), one has JJ = q. By recurrence, one derives from (12) the formula ! n n Y Y (13) G (χi , ψ) = J · G χi , ψ , i=1

i=1

where J is again Q an−1integer in Q (ζ) if the orders of the χi divide m. If one sets χ0 = i χi , one has, from (12) and (11): (14)

n Y

G (χi ψ) = qχ0 (−1) · J,

i=0

which shows that χ0 (−1)J depends symetrically on χ0 , χ1 , . . . , χn , thos being subject to the condition χ0 χ1 · · · χn = 1. For example, consider an automorphism τ of Q (ζ); if it maps ζ to ζ t , it maps χi into χti ; thus if (χt0 , . . . , χtn ) is a permutation of (χ0 , . . . , χn ), J will be invariant under τ . One can thus make sure that J belongs to a given subfield of Q (ζ). 8. Naturally, Gauss and his immediate successors until Kummer only worked with the Gauss sums relative to a prime field Fp and the corresponding sums J. It does not seem that Gauss himself saw the arithmetic importance of the integers J. However, he might have been struck by the fact that, for the cases m = 3 and m = 4, these integers give the decomposition of the rational prime numbers in the field Q (j), Q (i), where j 3 = 1, i4 = 1; this fact was known to him in another form (he expressed it using the ”periods”). Indeed let p ≡ 1 mod 3 (resp. mod 4); let χ be one of the two characters of order 3 (resp. of order 4) of F× p ; then J (χ, χ) is a prime factor of p in Q (j) (resp. Q (i)), and moreover satisfies important congruences. This is what Jacobi discovered; he even had the audacity, in 1827, to make Gauss share ([3 a]), which shows encouragement (with a point of condescension), but perhaps though, just as a little later in the matter of elliptic functions, that a young elephant tramped on the flower-beds. 9. Unlike Gauss, Jacobi recognized immediately the impact of the ”cyclotomic method”; this justifies the name of ”Jacobi sums”, that one nowadays gives to the integers J, although they already appear, as on has seen, in the secret papers of Gauss, and Cauchy introduced them and largely used them, from 1829 (independently from Jacobi), in some preliminary notes and especially in his great memoir of 1830 on the theory of numbers ([4]), that appeared with aditional notes in 1840. Cauchy was especially struck by the possibility (which follows from the remark at the end of no. 7) of constructing the sums J contained in a given quadratic extension of Q. For example, let l = 4n + 3 be prime; let r0 , . . . , rn be the quadratic residues mod l. Let p ≡ 1 mod l; let ri χ be a character of order l of Fp ; for 0 ≤ √i ≤
n, let χi = χ ; then (14) defines an integer J of the field k = Q −l , and one has JJ = pn−1 .

In addition, Cauchy determines the largest power pv of p dividing J; he can thus assert that 4pn−1−2v can be written in the form x2 + ly 2 . In modern language, this means that one has, in k, (J) = pv pn−1−2v , where p is one of the two prime factors of p. It is then a nontrivial result on the ideal class group of k, or, in the language of the time, on the group of classes of quadratic forms of discriminant −l; Jacobi, by the same reasoning (independently of Cauchy), even drew from it the correct conjecture on the number of these classes, some time before Dirichlet verified this conjecture in a celebrated work (largely anticipated by Gauss, as always in his ”secret papers”). 10. Jacobi was especially interested in the applications of the ”cyclotomic method” to the most burning problem in number theory at that time, the search for the nth power reciprocity laws for n > 2. On the subject of the law of biquadratic reciprocity, Gauss had just announced his results, in somewhat grandiloquent terms (”mysterium maxime reconditum”). Was he vexed to see Jacobi proclaiming that these result ”very simply and very easily” from his method? As always he never published his proof, which was based on very different principles. Furthermore, Jacobi did not either; his remained buried in his notes for the course in K¨onigsberg (1836-37); as for Jacobi’s statement, that same was obtained independently, a little later, by Eisenstein while still a student. For these cubic results ([5 a]), one can present the essential part as follows. In Z[j], 3 has the prime divisor ρ = j − 1. For any prime number π, relatively prime to 3, let q = N (π) = 3n + 1. For x relatively prime to π, on denotes the unique root of unity 1, j, or j 2 that is ≡ xn mod π by (x/π), and one extends this ”Legendre symbol” to a ”Jacobi symbol” by the rule (x/αβ) = (x/α) · (x/β) . Let p = 3v + 1 be a rational prime, and let π be one of its prime factors in Z[j]; one can, in a unique way, multiply π by a root (sixth) of 1 so that π becomes ”primary”, that is to say, ≡ 1 mod 3. Set χ(x) = (x/π) for x ∈ F× p ; it is a character × 2πix/p of order 3 of Fp . On Fp , one takes ψ(x) = e . Set G = G (χ, ψ), −1 2 J = J (χ, χ). One has then G (χ , ψ) = G, G = JG, G3 = pJ, GG = JJ = p, so then (15)

J=

p−1 X x=2

χ(x)χ(1 − x) ≡

p−1 X

xv (1 − x)v

mod π.

x=1

P But, for n 6≡ 0 mod (p − 1), one has p−1 xn ≡ 0 mod p; thus, J ≡ 0 1 mod π. As JJ = p, J/π is thus a sixth root of 1, which one determines as follows. One has set ρ = j − 1, hence ρ2 = −3j and j a = (1 + ρ)a ≡

1 + ρa mod 3. Set χ(x) = j i(x) ; one has i(xy) ≡ i(x) + i(y) mod 3, hence # " p−1 p−1 p−1 X X X i(x) i(1 − x) ≡ −1 + 2ρ J ≡p−2+ρ i(x) + 1

1

2

≡ −1

mod 3,

and hence J = −π. Now let σ be a prime in Z[j], relatively prime to 3p; let s = N (σ) = σσ; one has s ≡ 1 mod 3, χs = χ, thus X Gs ≡ χ(x)ψ(sx) ≡ χ (x)−1 G ≡ (s/π)−1 G mod σ. But in addition, if s = 3t + 1:

t Gs−1 = G3 = (−pπ)t ≡ −π 2 π/σ

mod σ.

One has (−1/σ) = (−1/σ)3 = 1, and also, by transport of structure, (π/σ) = (π/σ)−1 . As G is relatively prime to σ, the combination of the above relations gives then (s/π) = (π/s), which is the ”Eisenstein law”. If now one takes p0 to be a rational prime 6= p, ≡ 1 mod 3, and lets π 0 be a primary prime factor of p0 , one can, in the preceding, replace π, s, successively by π, p0 and by π 0 , p and combine the results. This gives first (π/π 0 )2 = (π 0 /π)2 , hence obviously (π/π 0 ) = (π 0 /π). One has thus all of the essentials of the cubic reciprocity law in Z[j]; the complementary results are easy to obtain. We also note immediately, with regard to the above example, a property of J of which Jacobi and his contemporaries attached much importance. For x ∈ F× p , one −1 −1 p−1−v has χ(x) = (x/π) = (x /π) ≡ x mod π, thus, from (15):   p−1 X 2v 2v 2v J≡ x (1 − x) ≡ − mod π. v x=1 This congruence, together with J ≡ 0 mod π,
completely determines J modulo p using the binomial coefficient 2v ; taking into account v commonplace inequalities, one can even say that it determines J, and therefore π, uniquely. 11. The example of no. 10 already contains the characteristic traits of ”cyclotomy”, that is to say of the theory of Gauss and Jacobi sums, such as it was developed in the 19th century. Initially, to use these sums, it is necessary to determine their decomposition into prime factors in the cyclotomic fields to which they belong. The solution for the sums of order 3 was seen above; it is

analogous for those of order 4; Jacobi also examined the sums of order 5, 8, and 12, by using the fact (which he noticed at the time) that the corresponding fields only have principal ideals. To go further, obviously, he needed the creation (by Kummer, starting in 1845) of ideal theory. What limited its impact for some time is that Kummer (who proceedeed by constructing explicitly the valuations in the fields in question) initially treated only the fields Q (ζ) with ζ l = 1, l an odd prime. One of his first triumphs was precisely obtaining the prime ideal decomposition of Gl in Z[l], for ζ l = 1, where p is prime, ≡ 1 mod l, and G is a Gauss sum of order l relative to Fp . A little later he realized (not for the Gauss sums, but, which comes to the same thing, for the Jacobi sums) that one could treat in the same way the finite fields Fq , those being presented as the residual fields of Z[ζ] modulo a prime ideal p (relatively prime to l) of degree > 1 (see [6]). 12. The decomposition into prime factors only determines the sums in question up to a unit; it was already insufficient for the sums of order 3 and 4; it is thus worse for the sums of order l, since there is then an infinite number of units in Z[ζ] by Dirichlet’s theorem (published in 1846). Also one desires additional precise details in the form of congruences. As in no. 11, these are of two kinds: (a) one, for the sums relative to Fp (resp. Fq with q = pn ) gives, not only the their order, but their principal parts in the places determined by the prime factors of p; (b) the other, even more important, concerns the local behavior of these sums in Q (ζ), or more generally at the places corresponding to the prime factors of m in Q (ζ) if they are sums of order m not prime, and if ζ m = 1. These questions led Kummer and Eisenstein to develop the very refined techniques of p-adic analysis, which unfortunately were later completely forgotten. 13. Finally, we again emphasize that, for Eisenstein and Kummer, cyclotomy appeared especially to be a means to tackle the problem of the reciprocity laws, within the framework where these were placed until Hilbert. For the mth power reciprocity law, Gauss’ example suggested placing it in the field Q (ζ) and no further, with ζ, as always, a primitive root of Z m = 1. For p relatively prime to m in Z[ζ], of norm q, and x relatively prime to p, one denotes by (x/p) the unique root ζ i that is ≡ x(q−1)/m mod p; one extends this ”Legendre symbol” to a ”Jacobi symbol” by the rule (x/ab) = (x/a) · (x/b). One then proposes

to obtain the expression, the most explicit possible, for (x/y) · (x/y)−1 , and also the ”supplementary laws” giving (x/p) when x is a unit or exactly divides m. Finally the hopes placed by Jacobi, Eisenstein, and Kummer in cyclotomy were only partially realized. It gives the ”Eisenstein law”, that is to say the value of (x/y)·(y/x)−1 when x (or y) is in Z; this is already not a small result. For m = 4, by a stroke of luck, one can get the complete statement of the biquadratic reciprocity law using the ”obvious” axiomatic properties of the symbol (x/y), that is to say, as one would say nowadays, by creating K-theory; it is undoubtedly what Jacobi did in his course at K¨onigsberg, and what Eisenstein did, who later applied his ideas to the K-theory of much more general problems. But more and more, until the end of his short life, Eisenstein devoted himself instead to the theory of elliptic functions in view of its arithmetic applications; it is this in particular that he gets his law of reciprocity from for m = 8. During the same time, Kummer, limiting himself once and for all to the lth power reciprocity laws for odd prime l (and even in fact for ”regular” l), used cyclotomy, with complete success, for the search for the ”supplementary laws”, but, to his great chagrin, had to record about 1853 that with these results and the Eisenstein law, it had provided all of which it was capable. 14. In 1890, Stickelberger revived and supplemented the results of Jacobi, Kummer, and Eisenstein which give the principal part of the Gauss and Eisenstein sums. We will summarize his work ([7]) in p-adic language, which doesn’t fundamentally change anything but allows us to be brief. Let p be prime, q = pn , and ω be a primitive root of W q−1 = 1; k = Qp (ω) is the unramified extension of Qp of degree n; on can identify Fq with Zp [ω]/(p), and Fp with Zp /(p). The automorphisms v of k over Qp map ω into ω p for 0 ≤ v < n, so that, if t designates the trace for k/Qp , one has:
n−1 (16) t ω i = ω i + ω ip + . . . + ω ip , and t (ω i ) is in Zp . Let ε be a primitive root of X p = 1 in an extension of k; for a ∈ Zp , on defines εa in the obvious way (by p-adic continuity, if one wants). Then x 7→ εt(x) , for x ∈ Zp [ω], defines, by passing to the quotient, a character ψ on the additive group Fq . In addition, the set of roots of X q = X in k is M = { 0, 1, ω, . . . , ω q−2 }; these are the multiplicative representatives of Fq in k. If thus, for x ∈ Zp [ω], one denotes by µx

the element of M that is ≡ x mod p, then x 7→ µx defines by passing to the quotient a character on F× q with values in k, and every character × on Fq with values in k is of the form x 7→ µ−a x . All of the Gauss sums relative to Fq , except for the trivial sum equal to −1, can thus be written in k (ε) in the form: X (17) ga = µ−a εt(µ) (0 < a < q − 1), µ

the sum being over µ ∈ M × = M − { 0 }. In k (ε), π = ε − 1 is a prime element, and one has, for all z ∈ Zp :   ∞ X z z i z , (18) ε = (1 + π) = π i 0 hence, for ga , the one has the convergent series   ∞ X X −a t(µ) i µ (19) ga = Aa,i π , Aa,i = . i µ i=0 P

We Q expressxρt(µ) using (16), and observe that the formal identity (1 + T ) (1 + T ) gives ! P  X Y x ρ  xρ = . i i P ρ ρ iρ =i

One obtains: (20)

Aa,i =

XX (iρ )

µ

µ

−a

Y µpρ  ρ

iρ

where 0 ≤ ρ ≤ n, where the second sum is over µ ∈P M × , and the first is over all systems of indices (i0 , . . . , in−1 ) such that iρ = i; for a given a, the smallest value of i for which Aa,i 6= 0 will be determined. The binomial coefficients that appear in the second P b member are polynomials in µ with coefficients in Q; moreover, µ has the value q − 1 or 0 according to whether b is or is not a multiple P of qρ − 1. Since 0 < a < q − 1, one can write a = aρ p with 0 ≤ aρ < p for 0 ≤ ρ < n. One has P P P (21) aρ = min ( jρ | jρ pρ ≡ a mod q − 1; jρ ≥ 0 (0 ≤ ρ < n)) , the minimum being attained only for j0 = a0 , . . . , jn−1 P = an−1 . Indeed, if one of the jρ , for example jλ , is > p, on can make jρ smaller by replacing jλ by jλ − p and jλ+1 (resp. j0 if λ P = n − 1) by jλ+1 + 1 (resp. j0 + 1); but if all of the jρ are < p, one has jρ pρ = a, hence jρ = aρ for all ρ.

xρ

=

That said, suppose Aa,i 6= 0. The second member of (20) must thus contain a term of degreeP≡ 0 mod q − 1; thisPimplies that there are integers iρP , jρ such that iρ = i, 0 ≤ jρ ≤ iρ , Pjρ pρ ≡ a mod q − 1, thus i ≥ aρ from (21). Furthermore, if i = aρ , these conditions imply that iρ = jρ = aρ for all ρ, which gives: Y (22) Aa,i = (q − 1) (aρ !)−1 . ρ

The principal part of ga is thus − ρ (π aρ /aρ !) . It is the definitive result on the question; one can say that it could already essentially be found in Kummer. Obviously one can deduce from this the principal part of the Jacobi sums, which Jacobi had already calculated in rather general cases ([3 b]). The methods of Stickelberger and of Kummer, and even undoubtedly of Jacobi, do not differ essentially from those that were just given. As this result gives the order of any Gauss sum (or Jacobi sum) relative to Fq , in any p-adic place, it obviously also contains the decomposition of all of these sums into prime factors. Q

15. All of this does not touch upon question (b) of no. 12, which is tied, on the one hand, to a part of the proof of the Eisenstein reciprocity law, and on the other hand to the property of the Jacobi sums which states that they define Hecke characters. We begin with the first, while placing ourselves at once in the most general case; we follow Eisenstein ([5 b]) freely, but rather closely. Let ζ be a primitive root of Z m = 1; let k = Q (ζ). Let p be a prime ideal (relatively prime to m) in k, of norm q = pn ; one identifies Fq with Z [ζ] /p; then (x/p) determines a character χ of order m over Fq . Let ε be a primitive root of X p = 1; let t be the trace for Fq /Fp ; x 7→ εt(x) is an additive character ψ on Fq . Set Φ (p) = (−1)m G (χ, ψ)m ; Φ does not depend on the choice of ε; one extends it to all ideals in k relatively prime to m by the rule Φ (ab) = Φ (a) Φ (b). Applying the result of no. 14 to G (χ, ψ) for the places of k determined by p and its conjugates, one easily finds the decomposition of the principal ideal (Φ (p)), then (Φ (a)), into prime factors; it is given by a symbolic power (23)

(Φ (a)) = aΘ

where Θ is an element of the group ring over the Galois group of k/Q, defined as follows. For all t ∈ (Z/mZ)× , let σt be the automorphism of k that maps ζ into ζ t . Then one has X −1 Θ= t · σ−t 0
(a result obtained by Kummer for prime m). In particular, one can apply (23) to a principal ideal a = (α), so that one can write Φ (α) = ε (α) · αΘ ,

(24)

where ε (α) is a unit of k. But in addition, the absolute value of the Gauss sums is given by (4); it follows immediately that |Φ (a)|2 = N (a)m ; taking into account (23) and (24), it follows that the unit ε (α), like all of its conjugates in k, has absolute value 1. Kronecker’s theorem shows that then ε (α) is a root of unity, of the form ±ζ i . Now let p0 be a prime ideal, relatively prime to m, of norm q 0 = m0 p = mv + 1. For p, χ, ψ as before, and p relatively prime to p0 , set G = G (χ, ψ); one has:  −1 X N (p0 ) q0 0 0 −1 G ≡ χ(x)ψ (q x) ≡ χ (q ) G ≡ G mod p0 . p But one also has (cf. the case m = 3 in no. 10):     (−1)m Φ (p) Φ (p) q 0 −1 m v G = (G ) ≡ ≡ p0 p0

mod p0 .

It follows that one has, whenever N (a), N (b) are relatively prime to each other and to m:   −1  Φ (a) N (b) = , a b since it is thus for a = p, b = p0 . We take a = (α), and apply (24), by observing that one has, by transport of structure, for tu ≡ 1 mod m, that is to say, σu = σt−1 :  σu    α α σu  α  u = σt = σt . b b b One obtains thus:   −1    ε (α) α N (b) = · . (25) α b N (b) 16. To get the Eisenstein law here, we now restrict ourselves (as Eisenstein did from the first) to the case where m is an odd prime number l. In this case, one has, with G = G (χψ) as always: X X (−G)l ≡ − χ (x)l ψ(lx) ≡ − ψ(lx) = 1 mod l, x6=0

x6=0

thus Φ (a) ≡ 1 mod l whatever a is (which answers question (b) in no. 12), and therefore ε (α) = ±1 whenever α is such that αΘ ≡ 1 mod (ζ − 1)2 ; it suffices for this that one has α ≡ x mod (ζ − 1)2 with x ∈ Z − lZ; Eisenstein then says that α is ”primary”. In modern language, these results also show that a 7→ Φ (a) is a ”Hecke character” (a ”Gr¨ossencharakter”) of conductor (ζ − 1)2 . If, in (25), one takes α to be ”primary”, and one takes b to be a prime ideal p of norm q = pn , one obtains (p/α)n = (α/p)n . But n divides l − 1, thus is relatively prime to l. One thus has (p/α) = (α/p), hence finally (a/α) = (α/a) whenever a is a rational integer relatively prime to l, and α is relatively prime to a and ”primary”. This is the Eisenstein law. 17. With regard to the more recent developments, we will be very brief. For reference, we recall that the Gauss sums appear among the local constant factors in the functional equations of the L functions; these factors are also called ”nombres radiciels” (”root-numbers”, ”Wurzelzahlen”), undoubtedly because of Hilbert, who a had a sort of genius for bad terminology, where it would have been sensible to name ”Wurzelzahl” that which before him had been named a ”Lagrange resolvent” , and ”Lagrangian Wurzelzahl” that which here has been called a Gauss sum. The constant factors of the functional equations for the Dirichlet L series appear for the first time in the calculation of L(1) by Dirichlet; this calculations is nothing other than, in substance, the verification of the functional equation connecting L(1) to L(0). Naturally, they reappear, in a more general form, in the functional equations for the Hecke and Artin L functions. They were the subject of considerable work by Dwork and by Langlands, finally supplemented by Deligne. Langlands highlighted the essential role played by these factors in representation theory. The author of these lines offers a medal (made out of chocolate) for he who proposes the best designation for the factors in question. 18. When one encounters algebraic numbers that, like all of their conjugates, have an absolute value of the form pn/2 with p prime, one is always tempted nowadays to wonder whether they are roots of zeta functions in characteristic p. It is indeed thus for the Gauss and Jacobi sums, as Hasse and Davenport realized in 1934 ([8]; cf. [9 a]); it is even on this occasion on which they discovered the important relation between Gauss sums which bears their name. In particular, the Jacobi sums of order m are roots (or poles, depending on the dimension) of

P the zeta functions of the varieties ai Xim = 0; in retrospect, one notes that particular cases, expressed in other language, were already known to Gauss, and that rather general cases are implicit in Kummer. We note in passing, as a matter of historical curiosity, that Gauss’ celebrated Tagebuch opens and closes with cyclotomy: it begins, dated March 30, 1796, by the division of the circle into 17 parts; it ends, on July 9 1814, with a note on the number of solutions of 1 = x2 +y 2 +x2 y 2 in Fp , connected to ”the theory of biquadratic residues” (thus to the ”periods” of order 4). As for the Hasse-Davenport relation, it links the Gauss sums of order m in Fq and in an extension FQ of Fq . Let Q = q N ; let t and n be the trace and the norm in FQ /Fq ; let G = G (χ, ψ) be a Gauss sum relative to Fq ; let G0 be the Gauss sum of G (χ ◦ n, ψ ◦ t) relative to FQ . Then one has −G0 = (−G)N . It should be said in passing, this shows once more that one has taken ”the wrong sign” in the usual notation for the Gauss sums. It is undoubtedly too late to rectify this mistake. 19. One can Papplymthe results cited in no. 18 on the zeta functions of the varieties ai Xi = 0 (and, we note in passing, all of the varieties that one can define as quotients of these latter by finite automorphism groups) to calculate the zeta functions of these same varieties over algebraic number fields. One finds that these functions are products of Hecke L functions, which amounts to saying that the Jacobi sums define Hecke characters in the cyclotomic fields. As one saw in no. 16, an important special case (relative to the sums (−G)l , where G is a Gauss sum of odd prime order l) forms the basis of Eisenstein’s proof of his reciprocity law. In fact, he gives there a more general result on the ”cyclotomic” Hecke characters in the abelian fields over Q (cf. [9, b, c]); naturally, it is the totally complex fields that are interesting from this point of view. Once these characters are obtained, one can propose to study the corresponding Hecke L functions, and notably their values L(s) for integral s. It is necessary to cite a remarkable result on this subject by Chowla and Selberg (see [10]); suitably interpreted, this shows that the value, at s = 1,√of the
L function defined by a certain ”cyclotomic” character over Q −n (for prime n ≡ 3 mod 4, it is just that which was defined as in Cauchy in no. 9) is simply expressed using π and the values of the function Γ(s) for s = a/n, 0 < a < n. One could undoubtedly go much further in this direction.

References [1] Lagrange. (a) R´eflexions sur la resolution alg´ebrique des ´equations, Nouveaux M´em. de l’Acad. R. des Sc. et B.-L. de Berlin, 1770-1771 = Oeuvres, vol. III, p. 332; (b) Trait´e de la r´esolution num´erique des ´equations. 2e ´ed., Paris 1808, Notes XIII-XIV = Oeuvres, vol. VIII, p. 295-367. [2] Gauss. (a) Disquisitiones arithmeticae. 1801 = Werke, vol. I; (b) Summatio serierum quorundam singularium. 1811 = Werke, vol. II, p. 11; (c) Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et ampliationes novae, 1818 = Werke, vol. II, p. 51; (d) Theoria residuorum biquadraticorum. Commentatio prime, 1828 = Werke, vol. II, p. 65; (e) Disquisitionum circa aequationes puras ulterior evolutio. Werke, vol. II, p. 243. [3] Jacobi. (a) Briefe an Gauss. Werke, vol. VII, p. 391-400; ¨ (b) Uber die Kreistheilung und ihre Anwendung auf die Zahlentheorie. Berl. Monatsber. 1837, p. 127 = Crelles J. Vol. 30 (1846), p. 166 = Werke, vol. VI, p. 254 [4] Cauchy. M´emoire sur la Th´eorie des Nombres. M´em. Ac. Sc. XVII (1840) = Oeuvres (I), vol. III. [5] Eisenstein. (a) Beweis des Reciprocit¨atssatzes f¨ ur die cubischen Reste in der Theorie der aus dritten Wurzeln der Einheit zusammengesetzten complexen Zahlen. Crelles J. 27 (1844), p. 189; (b) Beweis der allgemeinsten Reciprocit¨atsgesetze zwischen reellen und complexen Zahlen. Monatsber. d. k. Akad. d. Wiss. zu Berlin, 1850, p. 189. [6] Kummer. Ueber die Erg¨anzungss¨atze zu den allgemeinen Reciprocit¨ atsgesetzen. Crelles J. 44 (1851), p. 93. [7] Stickelberger, L. Ueber eine Verallgemeinerung der Kreistheilung. Math. Ann. 37 (1890), p.321. [8] Davenport, H. und H. Hasse. Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen F¨ allen, Crelles J. 172 (1935), p. 151. [9] Weil, A. (a) Numbers of solutions of equations in finite fields. Bull. Am. Math. Soc. 55 (1949), p. 197; (b) Jacobi sums as ”Gr¨ ossencharaktere”. Trans. Am. Math. Soc. 73 (1952), p. 487; (c) Sommes de Jacobi et caract`eres de Hecke. G¨ ott. Nachr. (forthcoming). [10] Selberg, A. and S. Chowla. On Epstein’s Zeta-Function. Crelles J. 227 (1967). p. 86.