具有三个奇素因子的判别式的二次数域的狭义类群

Narrow Class Groups of Quadratic Fields with Three Odd Prime Divisors in Discriminants

设F=Q(εp1p2p3)为二次数域,其中ε∈{±1,-2}和不同素数pi≡1 mod 4,i=1,2,3.给出F的狭义类群的4秩和8秩为1的充分必要条件.这些结果完全用同余关系、Legendre符号和4次剩余符号等来描绘,它们提供了有用的数值计算.

Let F=Q(εp_1p_2p_3) be quadratic number field,where ε∈{±1,-2} and distinct primes p_i≡1 mod 4,i=1,2,3.In this paper,the necessary and sufficient conditions for 4-rank and 8-rank of narrow class groups of F equal to 1 are given.Our characterizations are in terms of congruous relations,Legendre symbols,quartic residue symbols and so on.The results are very useful for numerical computations.