8-Ranks of Class Groups of Quadratic Number Fields and Their Densities  

作　　者：Qin YUE 

机构地区：[1]Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China

出　　处：《Acta Mathematica Sinica,English Series》2011年第7期1419-1434,共16页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China （Grant Nos. 10771100, 10971250）

摘　　要：For F = Q（√εpq）, ε∈ {±1,±2}, primes -p ≡ q ≡ 1 mod 4, we give the necessary and sufficient conditions for 8-ranks of narrow class groups of F equal to 1 or 2 such that we can calculate their densities. All results are stated in terms of congruence relations of p, q modulo 2^n, the quartic residue symbol （1/q）4 and binary quadratic forms such as q^h（-2p）/^4 = x^2 ＋ 2py^2 where h（-2p） is the class number of Q（√-2p）. The results are very useful for numerical computations.For F = Q（√εpq）, ε∈ {±1,±2}, primes -p ≡ q ≡ 1 mod 4, we give the necessary and sufficient conditions for 8-ranks of narrow class groups of F equal to 1 or 2 such that we can calculate their densities. All results are stated in terms of congruence relations of p, q modulo 2^n, the quartic residue symbol （1/q）4 and binary quadratic forms such as q^h（-2p）/^4 = x^2 ＋ 2py^2 where h（-2p） is the class number of Q（√-2p）. The results are very useful for numerical computations.

关 键 词：Class group unramified extension quartic residue 

分 类 号：O241[理学—计算数学] TP311.13[理学—数学]