Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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8-RANKS OF CLASS GROUPS OF IMAGINARY QUADRATIC NUMBER FIELDS AND THEIR DENSITIES

  • Jung, Hwan-Yup (Department of Mathematics Education Chungbuk National University) ;
  • Yue, Qin (Department of Mathematics Nanjing University of Aeronautics and Astronautics)
  • Received : 2010.07.13
  • Published : 2011.11.01
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Abstract

For imaginary quadratic number fields F = $\mathbb{Q}(\sqrt{{\varepsilon}p_1{\ldots}p_{t-1}})$, where ${\varepsilon}{\in}${-1,-2} and distinct primes $p_i{\equiv}1$ 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 = $\mathbb{Q}(\sqrt{{\varepsilon}p_1p_2)$, 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 $p_i$ modulo $2^n$, the quartic residue symbol $(\frac{p_1}{p_2})4$ and binary quadratic forms such as $p_2^{h+(2_{p_1})/4}=x^2-2p_1y^2$, where $h+(2p_1)$ is the narrow class number of $\mathbb{Q}(\sqrt{2p_1})$. The results are also very useful for numerical computations.

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References

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