Let F = Q（√-p1p2） be an imaginary quadratic field with distinct primes p1 = p2 = 1 mod 8 and the Legendre symbol （p1/p2） = 1. Then the 8-rank of the class group of F is equal to 2 if and only Pl if the following conditions hold： （1） The quartic residue symbols （p1/p2）4 = （p2/p1）4 = 1; （2） Either both p1 and p2 are represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=x^2-2p1y^2,x,y∈Z,or both p1 and p2 are not represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=ε（2x^2-p1y^2）,x,y∈Z,ε∈{±1},where h＋（2p1） is the narrow class number of Q（√2p1）,Moreover, we also generalize these results.
让 F 别是特征的一块地 2 和 3。让 f：M_( mn )(F)M_( mn )(F)是保存的一张添加剂地图{ 1 ， 2 ， T }-inverse,即 f (A)= f (A) f (B)~ Tf (A)，为任何 A 的 f (B) =f (B) f (A)~ Tf (B)，有 A=AB^TA 的 B M_( mn )(F)， B = BA^TB 。在这份报纸，我们给足够、必要的条件让 f 是如此的一张地图。
For quadratic number ?elds F = Q(√2p1 ···pt?1 ) with primes pj ≡ 1 mod 8, the authors study the class number and the norm of the fundamental unit of F. The resultsgeneralize nicely what has been familiar for the ?elds Q(√2p) with a prime p ≡ 1 mod 8, including density statements. And the results are stated in terms of the quadratic form x2 + 32y2 and illustrated in terms of graphs.
