The class number one problem for some non-abelian normal CM-fields

Louboutin, Stéphane; Okazaki, Ryotaro; Olivier, Michel

doi:10.1090/S0002-9947-97-01768-6

The class number one problem for some non-abelian normal CM-fields
HTML articles powered by AMS MathViewer

by Stéphane Louboutin, Ryotaro Okazaki and Michel Olivier PDF

Trans. Amer. Math. Soc. 349 (1997), 3657-3678 Request permission

Abstract:

Let $\textbf {N}$ be a non-abelian normal CM-field of degree $4p,$ $p$ any odd prime. Note that the Galois group of $\textbf {N}$ is either the dicyclic group of order $4p,$ or the dihedral group of order $4p.$ We prove that the (relative) class number of a dicyclic CM-field of degree $4p$ is always greater then one. Then, we determine all the dihedral CM-fields of degree $12$ with class number one: there are exactly nine such CM-fields.

References

Ezra Brown and Charles J. Parry, The imaginary bicyclic biquadratic fields with class-number $1$, J. Reine Angew. Math. 266 (1974), 118–120. MR 340219, DOI 10.1515/crll.1974.266.118
D. A. Buell, H. C. Williams, and K. S. Williams, On the imaginary bicyclic biquadratic fields with class-number $2$, Math. Comp. 31 (1977), no. 140, 1034–1042. MR 441914, DOI 10.1090/S0025-5718-1977-0441914-1
Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Jeffrey Hoffstein, Some analytic bounds for zeta functions and class numbers, Invent. Math. 55 (1979), no. 1, 37–47. MR 553994, DOI 10.1007/BF02139701
Kuniaki Horie, On a ratio between relative class numbers, Math. Z. 211 (1992), no. 3, 505–521. MR 1190225, DOI 10.1007/BF02571442
Christian U. Jensen and Noriko Yui, Polynomials with $D_{p}$ as Galois group, J. Number Theory 15 (1982), no. 3, 347–375. MR 680538, DOI 10.1016/0022-314X(82)90038-5
Stéphane Louboutin, Continued fractions and real quadratic fields, J. Number Theory 30 (1988), no. 2, 167–176. MR 961914, DOI 10.1016/0022-314X(88)90015-7
Stéphane Louboutin, Norme relative de l’unité fondamentale et $2$-rang du groupe des classes d’idéaux de certains corps biquadratiques, Acta Arith. 58 (1991), no. 3, 273–288 (French). MR 1121087, DOI 10.4064/aa-58-3-273-288
Stéphane Louboutin, Calcul des nombres de classes relatifs de certains corps de classes de Hilbert, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 4, 321–325 (French, with English and French summaries). MR 1289305
Stéphane Louboutin, Calcul du nombre de classes des corps de nombres, Pacific J. Math. 171 (1995), no. 2, 455–467 (French, with French summary). MR 1372239
Stéphane Louboutin, Lower bounds for relative class numbers of CM-fields, Proc. Amer. Math. Soc. 120 (1994), no. 2, 425–434. MR 1169041, DOI 10.1090/S0002-9939-1994-1169041-0
S. Louboutin. Majorations explicites du résidu au point $1$ des fonctions zêta des corps de nombres. to appear in the Journal of the Mathematical Society of Japan.
S. Louboutin. Determination of all quaternion octic CM-fields with class number $2.$ J. London Math. Soc. (2) 54 (1996), 227–238. .
Stéphane Louboutin and Ryotaro Okazaki, Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one, Acta Arith. 67 (1994), no. 1, 47–62. MR 1292520, DOI 10.4064/aa-67-1-47-62
Jacques Martinet, Sur l’arithmétique des extensions galoisiennes à groupe de Galois diédral d’ordre $2p$, Ann. Inst. Fourier (Grenoble) 19 (1969), no. fasc. 1, 1–80, ix (French, with English summary). MR 262210
H. L. Montgomery and P. J. Weinberger, Notes on small class numbers, Acta Arith. 24 (1973/74), 529–542. MR 357373, DOI 10.4064/aa-24-5-529-542
A. M. Odlyzko, Some analytic estimates of class numbers and discriminants, Invent. Math. 29 (1975), no. 3, 275–286. MR 376613, DOI 10.1007/BF01389854
M. Olivier. Table des corps cubiques réels de discriminant inférieur à $50000,$ avec base d’entiers, nombre de classes, régulateur et unités fondamentales. (January 1992), private communication.
Bennett Setzer, The determination of all imaginary, quartic, abelian number fields with class number $1$, Math. Comp. 35 (1980), no. 152, 1383–1386. MR 583516, DOI 10.1090/S0025-5718-1980-0583516-2
H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1–27. MR 222050
H. M. Stark, On complex quadratic fields wth class-number two, Math. Comp. 29 (1975), 289–302. MR 369313, DOI 10.1090/S0025-5718-1975-0369313-X
H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152. MR 342472, DOI 10.1007/BF01405166
Judith S. Sunley, Class numbers of totally imaginary quadratic extensions of totally real fields, Trans. Amer. Math. Soc. 175 (1973), 209–232. MR 311622, DOI 10.1090/S0002-9947-1973-0311622-9
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
Ken Yamamura, The determination of the imaginary abelian number fields with class number one, Math. Comp. 62 (1994), no. 206, 899–921. MR 1218347, DOI 10.1090/S0025-5718-1994-1218347-3