Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
System of polynomial equations wikipedia , lookup
Quartic function wikipedia , lookup
Factorization wikipedia , lookup
Field (mathematics) wikipedia , lookup
Eisenstein's criterion wikipedia , lookup
Cubic function wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
MATHEMATICS OF COMPUTATION VOLUME 46. NUMBER 174 APRIL 1986, PAGES 659-666 A Lower Bound for the Class Number of Certain Cubic Number Fields By Günter Lettl Abstract. Let ATbe a cyclic number field with generating polynomial i a— 3 ^ x3 —Y-x1 û+ 3 -=~-x- i and conductor m. We will derive a lower bound for the class number of these fields and list all such fields with prime conductor m = (a1 + 21)/A or m = (1 + 21b2)/A and small class number. 1. Introduction. Let hm denote the class number of the cyclotomic field Q(Çm), and A*, the class number of its maximal real subfield Q(cos(27r/w)). It is a well-known conjecture of Vandiver that p 1 hp holds for all primes p e P. This is a customary assumption for proving the second case of Fermat's Last Theorem (for more details see Washington [16]). Since hp grows slowly (hp = 1 for p < 163 with the use of the Generalized Riemann Hypothesis (GRH), van der Linden [10]), for no p with hp > 1 the exact value of hp is known without using GRH. Masley suggested that perhaps hp < p always holds, but a counterexample was found in [3], [12]. The class number of each real subfield of Q(f ) divides hp, and in this way one can find primes with hp > 1. Using the quadratic subfield, Ankeny, Chowla and Hasse [1] showed that hp > 1 if p belongs to certain quadratic sequences in N, and S.-D. Lang [9] and Takeuchi [15] found more such sequences. Similar results were obtained for h4p by Yokoi [17]. Using the cubic subfield of Q(f ), which has been thoroughly investigated (e.g., [2], [5], [8]), the theorem of the present paper yields the following results: If a is an odd integer, a > 23, andp = {a2 + 27)/4, a prime, then hp > 1. Ifb is an odd integer, b > 1, andp = (1 + 27A2)/4, aprime, then hp > 1. A conjecture about primes in quadratic sequences (Hardy and Wright [7, 1.2.8]) implies that there exist infinitely many primes p of each of these two forms, because one can write a2+ 27 /--*\2 + 3 ^z— +9 and I±™!_3(^lJL12 + ^±zl i Received November 6, 1984; revised July 15, 1985. 1980 Mathematics Subject Classification.Primary 12A30, 12A50. ©1986 American Mathematical Society 0025-5718/86 $1.00 + $.25 per page 659 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 660 GÜNTER LETTL 2. Class Number Bounds and Main Results. Let K be a cyclic cubic number field with conductor m and class number A. It is well known that m is the product of distinct primes, which are congruent to 1 mod (6), and of 9, if 3 ramifies in K. The class number A is congruent to 1 mod (3), if m is a prime or m = 9, and A is divisible by 3 otherwise. Set f(s) = L(s, x) ■L(s, x) for s£C, where x and x are the non trivial cubic Dirichlet characters modulo (m) belonging to K. Since the discriminant of K equals m2, the analytic class number formula yields (i) y> h = ^iM 4R ' where R is the regulator of K. Moser [11] showed that for prime conductors, A < m/3 holds, so cubic fields will never lead to a contradiction to Vandiver's conjecture. Our aim is to establish a lower bound for the class number of a special family of cubic fields and to list all fields of some special types with prime conductor and small class number. From a result of Stark [14] one can deduce /(l) > c/\ogm, where c is effectively computable, but this bound is not suited for our purposes. From the results of the next section we will obtain: (2) If K is a cyclic cubic number field with conductor m > 105, then /(l) > 0.023 • w" The harder problem is to find an upper bound for the regulator, which is only achieved for the following family of cyclic cubic fields. The polynomial (3) fa(X) = X3-^X2-^X-l, AENodd, is irreducible over Q, has discriminant D(fa) = ((a2 + 27)/4)2, and if e is a zero of fa, the other zeros are e' = -l/(e + 1) and e" = -(e + l)/e. Therefore, fa is a generating polynomial of a cyclic cubic field K with conductor m, and we define k e N by JD(fa) = (a2 + 27)/4 = km. We call the field K of type A, if k = 1, and of type B, if k = 27 and a = 21b with b e N odd, b # 1 (in this case we have m = (1 + 27b2)/4). It is well known that fields of type A or B have relatively large class numbers (see, for example, the tables of Gras [5]). Shanks [13] states that for cubic fields of type A with prime conductor "a rough mean value for A is given by A « 12w/35(log m)2 ". Lemma 1. Let K be a cyclic cubic field with generating polynomial fa, conductor m and regulator R. Then, (4) 4R<(\\o%D(fa))1 = (\o%(km))2. Proof of Lemma 1. Since the zeros e, e', e" of fa are units of K, we can estimate the regulator of K by R < Reg((e,e'}) =:R\ Choosing a - 3 + 4\[km ■cos(l/3 e =-ikm if R' # 0 (see Lemma 4.15 in [16]). • arctan(v/2T/a)) License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ,- THE CLASS NUMBER OF CERTAIN CUBIC NUMBER FIELDS 661 with the principal value of arctan, we obtain R' = det(log|a| \log|e'| logls'l log|e" = (log|e + 1|)2 - log|e + l|log|e| +(log|e|)2. Series expansions yield R' = \(\oêkm)2 - ^áM + _3_ + 0f tog_*m 2km 4km \ (km)1 and elementary calculus explicitly gives (4). With (2) and Lemma 1 we immediately obtain from (1): Let K be a cyclic cubic field with conductor m > 105 and with generating polynomial fa- Then, A > 0.023 (5) Theorem, m 0.946 (log km)2 (a) Let K be a cyclic cubic field of type A with prime conductor m. Then A < 16 holds onlyfor the followingvalues of m: A m 1 4 1,13,19, 37, 79, 97,139 163, 349,607,709, 937 313,877,1129,1567,1987,2557 1063 7 13 (b) Let K be a cyclic cubic field of type B with prime conductor m. Then A < 43 holds only for the following values of m: m 1 4 7 13 28 31 37 61, 331 547,1951 2437,3571 9241 4219, 25117 23497 8269 Proof of the Theorem. From (5) we obtain A > 14 for fields of type A with m ^ 169339, and A > 37.2 for fields of type B with m > 106. It is well known (see, e.g., Gras [4]) that primes q = -1 mod (3) divide the class number of a cyclic cubic field only with an even exponent. The table of class numbers of Shanks [13], and Table 1 below, complete the proof of the theorem. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use GÜNTER LETTL 662 Table 1 Class numbers of cyclic cubic fields of type B with prime conductor m < 106 1 + 2762 3 7 9 17 19 23 25 33 35 37 39 45 59 61 91 95 105 115 117 123 129 131 137 147 159 61 331 547 1951 2437 3571 4219 7351 8269 9241 10267 13669 23497 25117 55897 60919 74419 89269 92401 102121 112327 115837 126691 145861 170647 1 + 27¿2 1 1 4 = 22 4 = 22 7 7 173 185 189 191 193 199 28 = 22 49 = 72 37 13 49 = 72 109 31 28 = 22 1 133 = 7 • 19 193 205 221 227 231 235 243 259 261 297 299 301 688 = 24 43 211 532 22 7 ■19 303 307 347 361 367 604 = 22 151 148 = 22 37 97 652 = 22 163 628 = 22 157 305 341 371 373 383 202021 231019 241117 246247 251431 267307 283669 329677 347821 360187 372769 398581 452797 459817 595411 603457 611557 619711 627919 784897 812761 879667 909151 929077 939121 990151 316 79 343 1216 175 73 26 19 247 = 13-19 196 = 22 • 72 541 316 = 331 1732 = 553 = 1075 = 769 2257 = 2299 = 739 889 = 22 • 79 22 • 433 7-79 52 • 43 37-61 ll2 - 19 7-127 1156 22 ■172 1552 2" ■97 688 ' 24-43 769 688 24-43 787 1588 • 22 ■397 661 532 22 • 7 ■19 The class numbers of Table 1 were calculated with a " Sirius 1 Personal Computer", using the analytic class number formula (1). We also used that for fields of type B the roots of fa are already fundamental units, and therefore R = R' can be calculated with the explicit formula for e, given in the proof of Lemma 1. In the following way it can be proved that e is a fundamental unit: Let AT be a field of type B with generating polynomial fa, a = 21b and m = (1 + 27¿>2)/4. Hasse [8] investigated the arithmetic of cyclic cubic fields, using the Gauss sums of the corresponding Dirichlet characters. With Hasse's notation, every integer a G K can be written as a = [x, y] with x G Z, ye. Z[p], where p2 + p + 1 = 0, and x = y mod (1 - p). If a is a unit of K, N(a) = 1 implies x3 = 27 mod (m) and |jc| ^ 2^'myy (Satz 8, [8]). For the roots of f21b we have e = [(27A - 3)/2, 3/V3 ] and its conjugates. Since Godwin's conjecture about fundamental units holds for cyclic cubic fields with m > 9 (see Gras [6]), we have to show: There exists no unit a = [x, y] G K, a ¥= +1, with myy = \ tr(a — a')" < j tr(e - e')2 = 27w, where tr denotes the trace from K to Q. Suppose the contrary. Then x3 = 27mod (m) and \x\ < 2^2Jm imply x g {3,(27Z> - 3)/2, -(276 + 3)/2} for b > 7. Considering0 ■ x =ymod (1 - p) and yy < 27 yields only a few possibilities for y g Z[p], and one can check that for each License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 663 THE CLASS NUMBER OF CERTAIN CUBIC NUMBER FIELDS of these y, N(a) = 1 has no solution a ¥= 1. For small values of b, one can consult the table in [5]. In the same way, but with much less computation, one can prove that for k = 1 (type A) and k = 3 the roots of fa are also fundamental units. In these cases one has e = [(a- 3)/2, +1] with (a - 3)/2 = ±1 mod (3), and e = [(9b - 3)/2,/73], respectively. 3. A Lower Bound for L(l, x) ■L(l, x). Let m be the conductor of a cyclic cubic field K, x and x the nontrivial cubic Dirichlet characters modulo m associated with K, and f(s) = L(s, x)L(s, x)- To find a lower bound for /(l), we first need an upper bound for |/(*)| in a disk in C containing 1. Consider C = C(/t, p) = [s e C\\s — p\ < p) with 1 < n and ¡jl- 1 < p < ¡i, and set a0 = ¡u,- p. Let s = a + it e C For a > 0 we have the representation L(s,x)= m£^+s. n= l f^ldx " Jm withS(*,x)= X E x(«) l<n<x (see [16, p. 211]). The inequality of Pólya-Vinogradov [16, Lemma 11.8] states that |S(*>X)I < i/w • logm. For s g C(ja, p), the function \s\/a = l/cos(args) attains its maximum ¡i/ /ju2 - p2 if 5 is the point of contact of a tangent of C through 0. Combining these results, we obtain for every s G C{\i,p): (m \L(s, x) \ < 1 + / 1 /— /"°° "";r ^ + \s\\m ■logm 1 <-ml~°a + ^ logw - 1 dx • w05"°°. Since logx/ v^x is monotone decreasing for x > e2, we conclude that for m > m0 > (6) |/(5)| <q-m2-20" holds for all i g C(p, p), with _I / 1 log^o \2j M \1 - °o ' {^ p2 im Lemma 2. If K is a cyclic cubic number field with conductor m, thenf(l) > c6 ■m~Cl, with c6, c-j > 0 as given in the course of the proof. Furthermore, c7 can be made arbitrarily small. Proof of Lemma 2. The proof follows mainly Washington [16, pp. 212-214]. Let f(s) be the Riemann zeta function and ÇK(s) = ¡¡(s)f(s) the zeta function of the cyclic cubic field K with conductor m. If s = a + it g C, we have U*) = l+ 00 a E -7 fora>l, n= 2 with a„ > 0, and an > 1 if n is a cube. Developing f^ in a power series around jtt > 1 gives SM- Lbj(p-s)j, j-0 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 664 GÜNTER LETTL with 1 (7) b0 = £M> °° a í(3/i) > 1 and 6, = 77 E (log«)7 • -* > 0 for y > 1. The integral representation of f(s) for o > 0 yields roo |f(*)l< s- 1 + il/"00 s f- 1 A j aw = 5- 1 and |f(*)l< s< +1-1-Ê^r-r1 (-!•» du 1 5- n-1 " + Mi, 2 1 + ä1 1 Let C = C(jii, p), with ju - 1 < p < ¡ti, be the disk with center ¡i and radius p, and denote its boundary by dC. Using (6), we get for all s G 3C: 1 SM s-/(i)1 1/(1)1< ¿2 m 2-2o (8) i-i <lf(*)l•!/(*)! + with H +1 lí- Il c2 = c, • max se3C + \s \ ■min (MR)})- Since ÇK(s) - f(l)/(s - 1) is holomorphic in the whole complex plane, (8) holds for all s g C(ju, p). Computing the coefficients of /(I) (/i-l) ;-o\ 7+7h(M-^ with a Cauchy integral gives H^'-Ä) /(I) *;- (m-1) 7+1 ds £2 . w2-2o0 P' (,-M)y+1 For 0 < a < 1, the integral representation of f(s), and f(a) = \L(a, x)|2, show that ^k(°) < 0. So for any a with a0 < a < 1, and any y G R + with 1 < v, we have /(I) «-1 "-1 h! -c2 • w22-2o 2o° • V^ E If _ (ri)' 7TT -(^-«)J _ « w /(l) f(l)(p-a a —1 1 — a \ ju — 1 C • «r 2-2.JM-« a - an [-1-1 where E fy-ÎM~~a)y > c3 > 1. y-o From this inequality, we obtain 1 \ V /(l) > c3(l - a) M- 1 1 ju — a — Ct ■ m 2-2o0 . P2(l - «) (/i - a)(a License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ÍM-1 - o0) \ P THE CLASS NUMBER OF CERTAIN CUBIC NUMBER FIELDS 665 Choosing v = c4 ■logm + c5, with 2 - 2a0 C4 p log ¡u — a and log(, - Ik« - q0)+ lo^y~[ - ]°^y^i log- P H- a gives /(l) > c6 • m"'7, with C< CÁl aV-W C2 (,-a)(a-a0) [ p and c7 = (2-2a0)-log^/log^. Since c7 -» 0 for a -* 1, the proof of Lemma 2 is completed. Numerical computations show that for m0 = 105 good results can be obtained by choosing ft = 10, p = 9.9 and a = 0.975. With these values we obtain c2 = 10.8685 and v ~ 315. Using (7), and an > 1 for n a cube, we obtain the following estimations for c3: ["1-1 300 -, oo E Vm - a)J > U/0 + E -7 E -*((m- «)iog«)J 7= 0 /=i ■'■ n = 2 " W0 -, 300 , >í(3/0 + E7ÍE-T((M-«)3-log^)y A= 2 * > t , V j=\J- 1 í,.«.-» ((M - «)3 • log*)"1 *-»*^\ 301! 302 | ' 302-U-a)3-logfc]' where 7V0< e302/3(/x-<»)\yith the special values for p., p and a, and 7V0= 100, we obtain [cl-l 100 E ä,-(m- «)' > E *~2925 - 10"40 > 1-2175 = c3. 7=0 k=l These values yield c6 > 0.023 and, c1 < 0.054, and thus (2) is proved. Institut für Mathematik Karl-Franzens-Universität Halbärthgasse 1 A-8010 Graz Österreich, Austria 1. N. C. Ankeny, S. Chowla & H. Hasse, "On the class-number of the maximal real subfield of a cyclotomic field," J. ReineAngew.Math., v. 217,1965,pp. 217-220. 2. A. Châtelet, "Arithmétique des corps abéliens du troisième degré," Ann. Sei. Ecole Norm. Sup. (3), v. 63, 1946, pp. 109-160. 3. G. Cornell & L. C. Washington, "Class numbers of cyclotomic fields," J. Number Theory, v. 21, 1985, pp. 260-274. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 666 GÜNTER LETTL 4. M.-N. Gras, Sur le Nombre de Classes du Sous-Corps Cubique de Q'''' (p m 1 (3)), Thèse, Grenoble, 1971. 5. M.-N. Gras, "Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de Q," J. Reine Angew. Math., v. 277,1975, pp. 89-116. 6. M.-N. Gras, "Note a propos d'une conjecture de H. J. Godwin sur les unités des corps cubiques," Ann. Inst. Fourier (Grenoble), v. 30/4, 1980, pp. 1-6. 7. G. H. Hardy & E. M. WRIGHT,An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, New York, 1979. 8. H. Hasse, "Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubischen und biquadratischen Zahlkörpern," Abh. Deutsch. Akad. Wiss. Berlin Math.-Nat. KL, 1948, Nr. 2, 95 pp. 9. S.-D. Lang, "Note on the class-number of the maximal real subfield of a cyclotomic field", J. ReineAngew. Math., v. 290,1977,pp. 70-72. 10. F. J. van der Linden, "Class number computations of real abelian number fields," Math. Comp., v. 39,1982, pp. 693-707. U.C. MOSER, Sur le Nombre de Classes d'un Corps K Réel Cyclique de Conducteur Premier, deg K > 4, Séminaire Théorie des Nombres, Grenoble, 1980,17 pp. 12. E. Seah, L. C. Washington & H. C. Williams, "The calculation of a large cubic class number with an application to real cyclotomic fields", Math. Comp., v. 41, 1983, pp. 303-305. 13. D. Shanks, "The simplestcubic fields," Math. Comp.,v. 28,1974, pp. 1137-1152. 14. H. M. Stark, "Some effective cases of the Brauer-Siegel Theorem," Invent. Math., v. 23, 1974, pp. 135-152. 15. H. Takeuchi, "On the class-number of the maximal real subfield of a cyclotomic field," Canad. J. Math.,\. 33,1981,pp.55-58. 16. L. C. Washington, Introduction to CyclotomicFields, GTM 83, Springer, New York, 1982. 17. H. Yokoi, "On the diophantine equation x2 - py2 = ±Aq and the class number of real subfields of a cyclotomicfield," NagoyaMath. J., v. 91, 1983,pp. 151-161. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use