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For imaginary quadratic number fields F = Q(<수식>),where ε ∈ {-1, -2} and distinct primes pi ≡ 1 mod 4, we give conditions 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 = Q(<수식>), we com-pute 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 relations of pi modulo 2n, the quartic residue symbol (<수식>)4 and binary quadratic forms such as <수식>, where h+(2_(p1)) is the narrow class number of Q(<수식>). The results are also very useful for numerical computations.
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참고문헌 (16)
참고문헌 신청
E. Hecke / 1981 / Lecture on the Theory of Algebraic Numbers, GTM 77 / Springer-Verlag
F. Gerth III / 1982 / Counting certain number fields with prescibed l-class numbers / J. Reine Angew. Math 337 : 195 ~ 207
F. Gerth III / 1984 / Application of a character sum estimate to a 2-class number density / J. Number Theory 19 (2) : 239 ~ 247
F. Gerth III / 1984 / The 4-class ranks of quadratic fields / Invent. Math 77 (3) : 489 ~ 515
G. Hardy / 1979 / An Introduction to the Theory of Numbers, Fifth edition
F. Gerth III / 1982 / Counting certain number fields with prescibed l-class numbers / J. Reine Angew. Math 337 : 195 ~ 207
F. Gerth III / 1984 / Application of a character sum estimate to a 2-class number density / J. Number Theory 19 (2) : 239 ~ 247
F. Gerth III / 1984 / The 4-class ranks of quadratic fields / Invent. Math 77 (3) : 489 ~ 515
G. Hardy / 1979 / An Introduction to the Theory of Numbers, Fifth edition
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