Class numbers of imaginary abelian number fields
HTML articles powered by AMS MathViewer
- by Ku-Young Chang and Soun-Hi Kwon PDF
- Proc. Amer. Math. Soc. 128 (2000), 2517-2528 Request permission
Abstract:
Let $N$ be an imaginary abelian number field. We know that $h_{N}^{-}$, the relative class number of $N$, goes to infinity as $f_N$, the conductor of $N$, approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CM-fields $N$ with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of $N$. Second, we have proved in this paper that there are exactly 48 such fields.References
- Steven Arno, The imaginary quadratic fields of class number $4$, Acta Arith. 60 (1992), no. 4, 321–334. MR 1159349, DOI 10.4064/aa-60-4-321-334
- Ku-Young Chang and Soun-Hi Kwon, Class number problem for imaginary cyclic number fields, J. Number Theory 73 (1998), no. 2, 318–338. MR 1657972, DOI 10.1006/jnth.1998.2298
- Marie-Nicole Gras, Classes et unités des extensions cycliques réelles de degré $4$ de $\textbf {Q}$, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 1, xiv, 107–124 (French, with English summary). MR 526779
- Graziella Guerry, Sur la $2$-composante du groupe des classes de certaines extensions cycliques de degré $2^n$, J. Number Theory 53 (1995), no. 1, 159–172 (French, with English summary). MR 1344838, DOI 10.1006/jnth.1995.1084
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Kuniaki Horie and Mitsuko Horie, CM-fields and exponents of their ideal class groups, Acta Arith. 55 (1990), no. 2, 157–170. MR 1061636, DOI 10.4064/aa-55-2-157-170
- Kuniaki Horie, On a ratio between relative class numbers, Math. Z. 211 (1992), no. 3, 505–521. MR 1190225, DOI 10.1007/BF02571442
- Mikihito Hirabayashi and Ken-ichi Yoshino, Remarks on unit indices of imaginary abelian number fields, Manuscripta Math. 60 (1988), no. 4, 423–436. MR 933473, DOI 10.1007/BF01258662
- Mikihito Hirabayashi and Ken-ichi Yoshino, Remarks on unit indices of imaginary abelian number fields. II, Manuscripta Math. 64 (1989), no. 2, 235–251. MR 998489, DOI 10.1007/BF01160122
- Mikihito Hirabayashi and Ken-ichi Yoshino, Unit indices of imaginary abelian number fields of type $(2,2,2)$, J. Number Theory 34 (1990), no. 3, 346–361. MR 1049510, DOI 10.1016/0022-314X(90)90141-D
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, KANT V4, J. Symbolic Comput. 24 (1997), no. 3-4, 267–283. Computational algebra and number theory (London, 1993). MR 1484479, DOI 10.1006/jsco.1996.0126
- Franz Lemmermeyer, Ideal class groups of cyclotomic number fields. I, Acta Arith. 72 (1995), no. 4, 347–359. MR 1348202, DOI 10.4064/aa-72-4-347-359
- F. J. van der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), no. 160, 693–707. MR 669662, DOI 10.1090/S0025-5718-1982-0669662-5
- Stéphane Louboutin and Ryotaro Okazaki, The class number one problem for some non-abelian normal CM-fields of $2$-power degrees, Proc. London Math. Soc. (3) 76 (1998), no. 3, 523–548. MR 1616805, DOI 10.1112/S0024611598000318
- Stéphane Louboutin, Ryotaro Okazaki, and Michel Olivier, The class number one problem for some non-abelian normal CM-fields, Trans. Amer. Math. Soc. 349 (1997), no. 9, 3657–3678. MR 1390044, DOI 10.1090/S0002-9947-97-01768-6
- Stéphane Louboutin, CM-fields with cyclic ideal class groups of $2$-power orders, J. Number Theory 67 (1997), no. 1, 1–10. MR 1485424, DOI 10.1006/jnth.1997.2179
- 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
- Stéphane Louboutin, Upper bounds on $|L(1,\chi )|$ and applications, Canad. J. Math. 50 (1998), no. 4, 794–815. MR 1638619, DOI 10.4153/CJM-1998-042-2
- Stéphane Louboutin, Determination of all nonquadratic imaginary cyclic number fields of $2$-power degrees with ideal class groups of exponents $\leq 2$, Math. Comp. 64 (1995), no. 209, 323–340. MR 1248972, DOI 10.1090/S0025-5718-1995-1248972-6
- Stéphane Louboutin, Minoration au point $1$ des fonctions $L$ et détermination des corps sextiques abéliens totalement imaginaires principaux, Acta Arith. 62 (1992), no. 2, 109–124 (French). MR 1183984, DOI 10.4064/aa-62-2-109-124
- M. Ram Murty, An analogue of Artin’s conjecture for abelian extensions, J. Number Theory 18 (1984), no. 3, 241–248. MR 746861, DOI 10.1016/0022-314X(84)90059-3
- 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
- R. Okazaki, Inclusion of CM-fields and divisibility of relative class numbers, Preprint, 1996, Doshisha Univ.
- Young-Ho Park and Soun-Hi Kwon, Determination of all imaginary abelian sextic number fields with class number $\leq 11$, Acta Arith. 82 (1997), no. 1, 27–43. MR 1475764, DOI 10.4064/aa-82-1-27-43
- Young-Ho Park and Soun-Hi Kwon, Determination of all non-quadratic imaginary cyclic number fields of $2$-power degree with relative class number $\le 20$, Acta Arith. 83 (1998), no. 3, 211–223. MR 1611205, DOI 10.4064/aa-83-3-211-223
- H. M. Stark, Some effective cases of the Brauer-Siegel theorem, Invent. Math. 23 (1974), 135–152. MR 342472, DOI 10.1007/BF01405166
- 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
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
- 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
- K. Yamamura, Table of the imaginary abelian number fields with relative class number one and class number $>1$, Preprint(1999).
Additional Information
- Ku-Young Chang
- Affiliation: Department of Mathematics, Korea University, 136-701, Seoul, Korea
- Email: jang@semi.korea.ac.kr
- Soun-Hi Kwon
- Affiliation: Department of Mathematics Education, Korea University, 136-701, Seoul, Korea
- Email: shkwon@semi.korea.ac.kr
- Received by editor(s): May 1, 1998
- Published electronically: April 27, 2000
- Additional Notes: This research was supported by Grant BSRI-97-1408 from the Ministry of Education of Korea.
- Communicated by: David E. Rohrlich
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2517-2528
- MSC (1991): Primary 11R29; Secondary 11R20
- DOI: https://doi.org/10.1090/S0002-9939-00-05555-6
- MathSciNet review: 1707511