proceedings of the american mathematical society Volume 108, Number 4, April 1990

ON RATIONAL RECIPROCITY

CHARLES HELOU

(Communicated by William Adams)

Abstract. A method for deriving "rational" «th power reciprocity laws from general ones is described. It is applied in the cases n = 3,4, 8 , yielding results of von Lienen, Bürde, Williams.

Introduction Let n be a natural number greater than 1, and p, q two distinct prime numbers = 1 (mod n). Let Ç denote a primitive «th root of unity in C (the field of complex numbers), K = Q(£), with Q the field of rational numbers, and A = Z[Q, with Z the ring of rational integers. Since p = 1 (mod n), it splits completely in K ; so, if P is a prime ideal of A dividing p, the residue fields Z/pZ and A/P are naturally isomorphic, and P has absolute norm p . For a in A not divisible by P, the nth power residue symbol in K, modulo P, [a/P)n K , is then the unique «th root of unity satisfying the congruence

(a/P)nK = a{p-l)/n (modP). In particular, due to the residue fields isomorphism, for a in Z not divisible by p, (a/P)n K = 1 if and only if a is an «th power residue modulo p in Z. Thus, if Q is a prime ideal of A dividing q, an expression for the "rational inversion factor" (p/Q)n K • (q/P)"n K may be considered as a rational reciprocity law between p and q . 2. A rational reciprocity law In what follows, we assume that p and q are the norms in K/Q of prime elements v and w of A , respectively, and that a general «th power reciprocity law is given in K . This means that an expression is given for e(x,y) = (x/y)-(y/x)~i where x and y are relatively prime, and prime to «, elements of A, and (x/y) denotes the symbol (x/I)n K with / = y A. The latter symbol being

Received by the editors January 26, 1989 and, in revised form, May 5, 1989. Presented to the AMS meeting in Hoboken, NJ, October 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 11R04, 11A15.

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defined for any ideal I oí A prime to n and x , from the ones with / prime, by multiplicativity in / . Proposition. Let w = /(£), with f a polynomial with coefficients in Z and z a rational integer = Ç (mod v) (such a z exists and is unique modulo p, by the residue fields isomorphism). Then

(p/w) • (q/v)~ = e(p,w) • (m/v) where m is a rational integer such that m = Y[f(z ) "' (modp) k the product being extended to a set of positive representatives of the residue classes modulo n prime to n ; and for every k, k' is a positive integer such that kk' = 1 (mod «). Proof. Let G be the Galois group of K/Q. It is isomorphic to the multiplica- tive group (Z/«Z)* of congruence classes modulo « relatively prime to « : to the class k (mod «) corresponds the automorphism sk of K characterized by sk(Ç) = Ck • Since p (resp., q) is the product of the elements s(v) (resp., s(w)), for s ranging in G, we have, by the bilinearity of the power residue symbol, (1) (P/w) ■(q/v)~l = Y[(s(v)/w) ■(s(w)/v)~l. s Now, for every 5 in G,

(2) (s(v)/w)-(s(w)/v)~l = e(s(v) ,w) ■(w/s(v)) ■(s(w)/v)~l,

and, as follows easily from the definitions, (w/s(v)) —s(s~l(w)/v). Moreover, for every k (mod «) in (Z/«Z)*, sk(w) = f(Çk) = f(zk) (mod?;),

and, since sk = sk, with kk' = 1 (mod «),

5^ (w) = f(z ) (mod v).

Hence, (w/sk(v))-(sk(w)/v)~ = (f(z )/v) -(f(z )/v)~ . Therefore, in view of (2) and (1) and the bilinearity of e(x ,y), (3) (p/w)-(q/v)-1 =e(p,w)-Y[(f(zk')k/v).Y[(f(zk)n-l/v) k k with k (mod «) ranging in (Z/«Z)*, k and k' being positive. In the right- hand side of (3), one may exchange k and k' in the first product, then combine the two products into one, and eliminate the obvious «th power factors from the symbol. This yields the desired equality.

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Remark. In order to determine z and the power residue character of m mod- ulo v, in the proposition above, one may consider the norms x, of v over some subfields of K. On the one hand, these x, 's have v as common divisor, and so by linear combination and the solution of some congruence equation modulo v , one obtains z. On the other hand, upon replacing z by Ç in the expression of m , some of these x¡ 's appear in the product and allow us to sim- plify the resulting expression for m which is obtained by rechanging £ into z . As an interpretation of these x¡ 's, note that their norms over Q provide rep- resentations of p by some norm forms. Finally, the expression of (m/v) may, in some cases, be simplified further by using the power reduction property (if d is a positive divisor of « , then (x/y)n K = (x/y)n,d K) and the norm property (if x is in a subfield E of K containing Ç and N is the norm in K/E, then (x/y)n K = (x/Ny)n E). These follow from the fundamental properties of the symbol (for which, see [3], Chapter 14).

3. Examples (i) « = 3 . Let v = a + bÇ, with Ç a primitive third root of unity, a and b in Z. So p = Nv = a -ab + b , and a, b are prime to p . Let b' be an inverse of b modulo p, in Z. Let w = c + dÇ = f(Q. Then, by the proposition above, with z = -ab' (mod p), (4) (p/w).(q/v)-l=e(p,w)-(f(z2)/v). By the cubic reciprocity law ([2], p. 73), e(p,w) = e(v ,w)-e(v ,w) = 1 (where v is the complex conjugate of v ), provided that v and w are primary, i.e. a, c = 2 (mod 3) and b, d = 0 (mod 3). Note also that z2 = -a'b (mod p), with a! in Z such that aa' = 1 (mod p). Hence the right-hand side of (4) is ((c - da'b)/v) = (a (ac - bd)/v). The resulting expression was given by von Lienen in [4]. It can be simplified by again using the cubic reciprocity law which yields (a/v) = £(a+1)/3. We thus obtained Corollary 1. Let p ,q be two prime numbers = 1 (mod 3), p = a - ab + b , q — c - cd + d , with a,c = 2 (mod 3) and b,d = 0 (mod 3), in Z. Let v = a + bÇ and w — c + dC,, in K = Q(Q , with Ç a primitive cubic root of unity. Then (p/w)iK ■(q/v)-]K = r(a+1)/3 • ((ac - bd)/v)3K. (ii) « = 4. Let v = a + bi, with i a primitive fourth root of unity, and a, b in Z. So p = Nv = a + b . Let a and b' be inverses of a and b modulo p, respectively, in Z. Then, by the proposition above, with z = -ab' = a'b (mod p), and w = c + di = f(i), (5) (p/w)-(q/v)-X =e(p,w)-(f(zi)/vf. By the biquadratic reciprocity law ([2], p. 138), e(p ,w) = (_1)«p-')/2)-(c-i)/2 = 1 , provided that w is primary, i.e. c is odd and d = c - I (mod 4). Note

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also that z = -a'b = ab' (mod p). Thus, using the power reduction and the norm properties of the symbol, the right-hand side of (5) is ((c-da'b)/v)4 K = (a/p)2 q • ((ac - bd)/p)2 Q . Moreover, provided a is odd, the quadratic reci- procity law for the Jacobi symbol yields (a/p)2Q = 1 . Alternatively, with ab' substituted for z , we would have obtained, for the right-hand side of (5), (b2/v)4K ■((bc + ad)/p)2Q; and (b2/v)4K = (-a2/v)4K = (-\/v),K = (-1) . Finally, note that the left-hand side of (5) and both expressions obtained for the right-hand side are invariant if v or w is changed into its opposite (since -1 is a quadratic residue mod p ). So it is enough to assume that a and c are odd, and then either v or —v (resp., w or —w ) is primary. Thus 2 2 Corollary 2. Let p ,q be two prime numbers = 1 (mod 4), p = a + b , q = c 2 + d 2 , with a,b,c,d in Z and a,c odd. Let v = a + bi and w = c + di in K = Q(i). Then

(PM4

In particular, one obtains Burde's law [1] upon replacing v by its complex conjugate v = a- bi in the above, since (q/v)4 K = (q/v)^ K . (iii) « = 8. Since K = Q(Ç), with Ç a primitive eighth root of unity, has class number 1, there always exist prime elements v and w in A = Z[£] with norms, in K/Q,p and q, respectively. Moreover, v and w may be chosen primary, i.e. = 1 (mod 2(Ç - 1)). The Galois group of K/Q consists of sk , for k = 1,3,5,7, where sk(Ç) = Ç . So, K has two imaginary quadratic subfields £■, = Q(i) and E2 = Q(/\/2), fixed fields of {s¡ ,s5} and {5, ,s3}, respectively. Let xx = vs5(v) = fl, + b{i and x2 = vs3(v) = a2 + b2(Ç + Ç ) be the norms of v over E{ and E2, respectively, where ax ,bl,a2,b2 are in Z (here Ç = / ; Ç + Ç = i\/2). Then, taking norms over Q, we get p = 2 2 2 2 a, + è, = a2 + 2b2. Also, v being a common divisor of x, and x2, we have èjjc2-Cè2xi = bla2 + b2(bl-al)Ç = 0 (mod v). Thus, the value of Ç (mod v) is z = a2b\b'2(ax - bx)' (mod/?), where, for r in Z not divisible by /?, r denotes an inverse of r modulo p . Let w = /(C), with / in Z[X]. By the proposition above and the power reduction and the norm properties of the symbol, we have

(6) (P/W)$,K • (llV\l,K - e(P>W) • (/(zVU5)2/U7)3A*l)4,£,- By Eisenstein's octic reciprocity law ([2], p. 616), provided w is primary, e(p,w) = (-lfpi-1)/S)^-^ = l. Now, let y, = f(Of(Ç5) = c, + dxi and v2 = /(C)/(£3) = c2 + ¿2(Ç + £3) 2 2 2 2 be the norms of w over £, and E2, respectively; so q = c, + úf, = c2 + 2d2 , with cx,dx,c2,d2 in Z. We have

/(c3)/(c7) = ^i ) = cx - dxi ; /(c5)/(c7) = *5(y2)= c2 - ¿2(c + c3).

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So, in virtue of the residue fields isomorphism (and noting that i = C ), f(z3)f(z7) = cx - dxz2 (mod p); f(z5)f(z7) = c2 - d2(z + z3) (mod p). Hence, f(z3)f(z5)2f(z7)3 = (cx - dxz2)(c2 - d2(z + z3))2 (mod p). It follows that the right-hand side of (6) is equal to uxu2 where ux = ((c, - dxz )/x,)4 E and u2 — ((c2 - d2(z + z ))/p)2 Q . Moreover, it is easily seen that z = axbx = -axb\ (mod/?) and z ^r z =-a2b'2 (mod/?). Hence

"l = (V-*l)4,í-, • ((fllCl - Vl)/-*l)4,£, = (¿l/-Xl)7Í,-((ulú?l+¿lí:i)Ml)4,£, and u2 = (b2/p)2Q ■((a2d2 + b2c2)/p)2Q. Also, provided v is primary in A , its norm x, is primary in Z[i], and then, 2 2 since p = ûj + ¿, = 1 (mod 8) and ¿>, = a, — 1 (mod 4), we have ax = 1 (mod 4). So, by the biquadratic reciprocity law,

K/*l)4,£, = (Vai)4,£, = (//öl)4,£l = (-1)(P"1,/8-

Consequently, (bx/xx)4 E = (iax/xx)4 E = 1. On the other hand, using the fact that 2 is a quadratic residue modulo p and the quadratic reciprocity law, we get (b2/p)2 Q = 1 . Hence

»i = (-l)(p~1)/8 • ((axcx - bxdx)/xx)4E¡ = ((axdx + bxcx)/xx)4E¡ and u2 = ((a2d2 + b2c2)/p)2Ç}. Finally, note that ux and u2 are unchanged if any of xx ,x2,yx,y2 is replaced by its opposite; if x, is chosen a priori, having norm over Q equal to p and with a, odd, then a primary prime divisor v of p in A has for norm, over Ex, x, or -xx ; and the left-hand side of (6) is invariant if v (or w ) is replaced by any of its associates. Similar remarks apply to yx, and also to x2 and to y2. Thus

2 2 Corollary 3. Let p and q be two prime numbers = 1 (mod 8), p — ax + bx = 2 2 2 2 2 2 a2 + 2b2 and q = cx + dx = c2 + 2d2, with ax,bx,a2,b2,cx,dx,c2, d2 in Z and ax,a2,cx ,c2 odd. Let v (resp., w) be a common prime divisor in Z[Ç] of ax + bxi and a2 + b2i\¡2 (resp., of c, + dxi and c2 + d2i\¡2), where C " a primitive eighth root of unity and i = Ç . Also K = Q(Ç), E = Q(i). Then

(PM»jc ■(QM*!k = (-1)(""1)/8 • ((axcx - bxdx)/(ax + bxi))4E •((a2d2 + b2c2)/p)2Q = ((axdx +bxcx)/(ax + bxi))4E- ((a2d2 + b2c2)/p)2X}. In particular, one obtains Williams' law [5] upon replacing w by its complex conjugate and exchanging the roles of p and q .

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References

1. K. Bürde, Ein rationales biquadratisches Reziprozitätsgesetz, J. Reine Angew. Math. 235 (1969), 175-184. 2. G. Eisenstein, Mathematische Werke I and II, Chelsea, 1975. 3. K. Ireland and M. Rosen, A classical introduction to modern number theory, Springer, Berlin, 1982. 4. H. von Lienen, Reelle kubische und biquadratische Legendre-Symbole, J. Reine Angew. Math. 305(1979), 140-154. 5. K. S. Williams, A rational octic reciprocity law, Pacific J. Math. 63 (1976), 563-570.

Department of Mathematics, Pennsylvania State University, Delaware County Campus, 25 Yearsley Road, Media, Pennsylvania 19063

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