no. 2
The imaginary abelian number fields with class numbers equal to their genus class numbers
Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 2, pp. 349-365.

Titre français : Sur les corps abéliens dont le nombre de classes est égal au nombre de genres. Nous savons qu'il n'existe qu'un nombre fini de corps abéliens imaginaires pour lesquels le nombre de classes est égal au nombre de genres. Ceux de ces corps qui sont cycliques et non quadratiques ont été classés dans [Lou2,4] et dans [CK]. Dans cet article, nous déterminons tous les corps abéliens non cycliques dont le nombre de classes est égal au nombre de genres. Cela achève la classification des corps abéliens possédant une classe par genre, sauf dans le cas des corps quadratiques imaginaires.

We know that there exist only finitely many imaginary abelian number fields with class numbers equal to their genus class numbers. Such non-quadratic cyclic number fields are completely determined in [Lou2,4] and [CK]. In this paper we determine all non-cyclic abelian number fields with class numbers equal to their genus class numbers, thus the one class in each genus problem is solved, except for the imaginary quadratic number fields.

@article{JTNB_2000__12_2_349_0,
     author = {Chang, Ku-Young and Kwon, Soun-Hi},
     title = {The imaginary abelian number fields with class numbers equal to their genus class numbers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {349--365},
     publisher = {Universit\'e Bordeaux I},
     volume = {12},
     number = {2},
     year = {2000},
     mrnumber = {1823189},
     zbl = {0972.11107},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2000__12_2_349_0/}
}
TY  - JOUR
AU  - Chang, Ku-Young
AU  - Kwon, Soun-Hi
TI  - The imaginary abelian number fields with class numbers equal to their genus class numbers
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2000
SP  - 349
EP  - 365
VL  - 12
IS  - 2
PB  - Université Bordeaux I
UR  - http://www.numdam.org/item/JTNB_2000__12_2_349_0/
LA  - en
ID  - JTNB_2000__12_2_349_0
ER  - 
%0 Journal Article
%A Chang, Ku-Young
%A Kwon, Soun-Hi
%T The imaginary abelian number fields with class numbers equal to their genus class numbers
%J Journal de théorie des nombres de Bordeaux
%D 2000
%P 349-365
%V 12
%N 2
%I Université Bordeaux I
%U http://www.numdam.org/item/JTNB_2000__12_2_349_0/
%G en
%F JTNB_2000__12_2_349_0
Chang, Ku-Young; Kwon, Soun-Hi. The imaginary abelian number fields with class numbers equal to their genus class numbers. Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 2, pp. 349-365. http://www.numdam.org/item/JTNB_2000__12_2_349_0/

[A] S. Arno, The imaginary quadratic fields of class number 4. Acta Arith. 60 (1992), 321-334. | EuDML | MR | Zbl

[CK] K.-Y. Chang, S.-H. Kwon, On the imaginary cyclic number fields. J. Number Theory 73 (1998), 318-338. | MR | Zbl

[G] M.-N. Gras, Classes et unités des extensions cycliques réelles de degré 4 de Q. Ann. Inst. Fourier (Grenoble) 29 (1979), 107-124. | EuDML | Numdam | MR | Zbl

[H] H. Hasse, Über die Klassenzahl abelscher Zahlkörper. Academie-Verlag, Berlin, 1952. Reprinted with an introduction by J. Martinet: Springer-Verlag, Berlin (1985). | MR | Zbl

[HY1] M. Hirabayashi, K. Yoshino, Remarks on unit indices of imaginary abelian number fields. Manuscripta Math. 60 (1988), 423-436. | EuDML | MR | Zbl

[HY2] M. Hirabayashi, K. Yoshino, Remarks on unit indices of imaginary abelian number fields II. Mamuscripta Math. 64 (1989), 235-251. | EuDML | MR | Zbl

[HY3] M. Hirabayashi, K. Yoshino, Unit indices of imaginary abelian number fields of type (2,2,2). J. Number Theory 34 (1990), 346-361. | MR | Zbl

[K] T. Kubota, Über den bizyklischen biquadratischen Zahlkörper. Nagoya Math. J. 10 (1956), 65-85. | MR | Zbl

[KT] M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, K. Wildanger, KANT V4 J. Symbolic Comp. 24 (1997), 267-283. | MR | Zbl

[L] F. Van Der Linden, Class number computations of real abelian number fields. Math. Comp. 39 160 (1982), 693-707. | MR | Zbl

[Lou1] S. Louboutin, A finiteness theorem for imaginary abelian number fields. Manuscripta Math. 91 (1996), 343-352. | EuDML | MR | Zbl

[Lou2] S. Louboutin, The nonquadratic imaginary cyclic fields of 2-power degrees with class numbers equal to their genus class numbers. Proc. Amer. Math. Soc. 27 (1999), 355-361. | MR | Zbl

[Lou3] S. Louboutin, Minoration au point 1 des fonctions L et determination des sextiques abéliens totalement imaginaires principaux. Acta Arith. 62 (1992), 535-540. | EuDML | MR | Zbl

[Lou4] S. Louboutin, The imaginary cyclic sextic fields with class numbers equal to their genus class numbers. Colloquium Math. 75 (1998), 205-212. | EuDML | MR | Zbl

[Lou5] S. Louboutin, Minorations (sous l'hypothese de Riemann généralisée) des nombres de classes des corps quadratiques imaginaires. Application, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 795-800. | MR | Zbl

[M] I. Miyada, On imaginary abelian number fields of type (2,2,..., 2) with one class in each genus. Manuscripta Math. 88 (1995), 535-540. | EuDML | MR | Zbl

[Ma] J.M. Masley, Class numbers of real cyclic number fields with small conductor. Compositio Math. 37 (1978), 297-319. | EuDML | Numdam | MR | Zbl

[Mä] S. Mäki, The determination of units in real cyclic sextic fields. Lecture notes in Math. 797, Springer-Verlag, Berlin and New York (1980). | MR | Zbl

[P] C. Batut, D. Bernardi, H. Cohen,M. Olivier, PARI-GP, version 1. 38.

[S1] H.M. Stark, A complete determination of the complex quadratic fields of class number one. Michigan Math. J. 14 (1967), 1-27. | MR | Zbl

[S2] H.M. Stark, On complex quadratic fields with class number two. Math. Comp. 29 (1975), 289-302. | MR | Zbl

[W] L.C. Washington, Introduction to cyclotomic fields. GTM 83, 2nd Ed., Springer-Verlag (1996). | MR | Zbl

[We] P.J. Weinberger, Exponents of the class groups of complex quadratic fields. Acta Arith. 22 (1973), 117-124. | EuDML | MR | Zbl

[Y] K. Yamamura, The determination of imaginary abelian number fields with class number one. Math. Comp. 62 (1994), 899-921. | MR | Zbl

[YH1l K. Yoshino, M. Hirabayashi, On the relative class number of the imaginary abelian number field I. Memoirs of the College of Liberal Arts, Kanazawa Medical University, Vol. 9 (1981). | Zbl

[YH2] K. Yoshino, M. Hirabayashi, On the relative class number of the imaginary abelian number field II. Memoirs of the College of Liberal Arts, Kanazawa Medical University, Vol. 10 (1982) . | Zbl