On The Class Group And Class Number Of Non-full Real Number Cubic Field

The research on class group and class number in the number field is based on ideal theories and methods,and the research results play an important role in the number field theory.Class number is an important invariant of the number field K,which can be regarded as a measure of the gap between OK and the principal ideal domain,which can reflect the stability of the element under the unique decomposition rule,and the class group can be regarded as a measure of the process of logarithmic expansion to ideal.The cubic field is a very important research object in algebraic number theory,and the study of its class number and fundamental unit is the central topic in algebraic number theory.This paper studies the class group and class number of some special cubic field,as well as the estimation of the upper bound of the fundamental unit.For non-full real number cubic field K=Q(?),? is the root of irreducible polynomial f(x)=x3+ax+b,where(1<a<9,b=±5,±7).First,this paper uses the existing methods and conclusions to find the discriminant and integral basis of the cubic field,In particular,to solve the case where the discriminant of the element ? for the expansion K/Q contains several square factors,the structure of the set A={[?]|?|p?M(K)}is determined.On this basis,using the prime ideal decomposition feature to examine the multiplicative relationship between the elements in A,certain class group and class number of the non-full real number cubic field are determined.It is more difficult to find the fundamental unit of the cubic field,this paper uses analytical methods to estimate the effective upper bound of the fundamental unit of nonfull real number cubic field,to further explore the application of the upper bound,a proof of the fundamental unit of a class of cubic field is given.Finally,the existence of fundamental unit in a class of companion elements is discussed,it is proved that the class group contains elements of order 2,a method is proposed to determine the order of the element by using the relationship between the fundamental unit and the companion element.