8-ranks Of Class Groups For Imaginary Quadratic Number Fields

The class group of a number field is an important subject in algebraic number theory. This paper mainly studies the structure of 2-Sylow subgroups of class groups for imaginary quadratic number fields.By genus theory, Rédei criteria, Gauss theory and Legendre theory, this paper computes the 8-ranks of class groups for imaginary quadratic number fields, namely in decompositions of their 2-Sylow subgroups into direct summands of cyclic subgroups, 8-ranks of class groups are the number of all direct summands whose orders are divible by 8 .This paper mainly investigates 8-ranks of class groups for imaginary quadratic number fields, where each prime divisor of their discriminant is congruous 1 modulo 4.When imaginaray quadratic fields have either two or three odd prime divisors in their discriminants, we give necessary and sufficient conditions for 8-ranks of class groups being 1 or 2 completely. Moreover we generalize above results into a lot of odd prime divisors in their discriminants. All results are stated in terms of congruence relations, Legendre symbol and the quartic residue symbol. They are very useful for numberical computations. Furthermore, we also generalize the remarkable reciprocity law discovered by K.Burde and our results will be simpler, and apply for computing 8-ranks of class groups.