The Ideal Class Group Of An Imaginary Quadratic Field And Ono Number

The finite extension field K of Q is called algebraic number field, or number field for short. This is the object of basic research of algebraic number theory. If the degree of K is 2, and K(?)R, then K is also called an imaginary quadratic field. In this paper, we study the ideal class group of an imaginary quadratic field and Ono number. Let D be the discriminant of Q((?)), to for its number of ramified primes, pp for its Ono number, qD for the smallest prime number which splits completely in Q((?)).In [1], the author gives a complete list of the Ono number equal to 3 of an imaginary quadratic field under the Extended Riemann Hypothesis, and proves that an imaginary quadratic field Q((?)) has the ideal class group isomorphic to Z/2Z(?)/2Z if and only if the Ono number of Q((?)) is 3 and Q((?)) has exactly 3 ramified primes. The main purpose of this paper is to study the Ono number equal to 4 of an imaginary quadratic field. In this paper, we use the theory of Gauss and Sasaki’s inequality to find the neces-sary condition for the ideal class group of imaginary quadratic field isomorphic to Z/4Z: tD=2, pD=3 or 4. When D meets PD=4, we use the inequalities relating to D to get |D|<3.39×1015, then we search out the D which meets pD=4 and-2×105<D<0 by GP. Thus, we have the following conclusion:the Ono number of an imaginary quadratic field is not only related to the class number, but also to the structure of the ideal class group, which has a very rich nature.