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Keywords:
imaginary abelian number field; relative class number; determinant; class number formula
imaginary abelian number field; relative class number; determinant; class number formula
Summary:
We give a new formula for the relative class number of an imaginary abelian number field $K$ by means of determinant with elements being integers of a cyclotomic field generated by the values of an odd Dirichlet character associated to $K$. We prove it by a specialization of determinant formula of Hasse.
References:
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[2] Hasse, H.: Über die Klassenzahl abelscher Zahlkörper. 1952, Akademie-Verlag, Berlin, Reprinted with an introduction by J. Martinet, Springer Verlag, Berlin (1985). MR 0842666 | Zbl 0046.26003
[3] Hirabayashi, M., Yoshino, K.: Remarks on unit indices of imaginary abelian number fields. Manuscripta math., 60, 1988, 423-436, DOI 10.1007/BF01258662 | MR 0933473 | Zbl 0654.12002
[4] Washington, L.C.: Introduction to Cyclotomic Fields, 2nd edition. 1997, Springer Verlag, Berlin, MR 1421575
[5] Yamamura, K.: Bibliography on determinantal expressions of relative class numbers of imaginary abelian number fields. Number Theory. Dreaming in Dreams. Proceedings of the 5th China-Japan Seminar, 2010, 244-250, World Sci. Publ., Hackensack, MR 2798466 | Zbl 1202.11001
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