8-RANKS OF CLASS GROUPS OF IMAGINARY QUADRATIC NUMBER FIELDS AND THEIR DENSITIES

- Journal title : Journal of the Korean Mathematical Society
- Volume 48, Issue 6, 2011, pp.1249-1268
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2011.48.6.1249

Title & Authors

8-RANKS OF CLASS GROUPS OF IMAGINARY QUADRATIC NUMBER FIELDS AND THEIR DENSITIES

Jung, Hwan-Yup; Yue, Qin;

Jung, Hwan-Yup; Yue, Qin;

Abstract

For imaginary quadratic number fields F = , where {-1,-2} and distinct primes mod 4, we give condition of 8-ranks of class groups C(F) of F equal to 1 or 2 provided that 4-ranks of C(F) are at most equal to 2. Especially for F = , we compute densities of 8-ranks of C(F) equal to 1 or 2 in all such imaginary quadratic fields F. The results are stated in terms of congruence relation of modulo , the quartic residue symbol and binary quadratic forms such as , where is the narrow class number of . The results are also very useful for numerical computations.

Keywords

class group;unramified extension;quartic residue;density;

Language

English

Cited by

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