On The Class Group And Class Number Of Totally Real Cubic Field

The basic object of study of algebraic number theory is algebraic number field,class group and class number of number field is a center research in algebraic number theory,it’s study results have important guiding significance in decomposition and use of the number of number field.When a prime number can’t be decomposed in the field of rational number,it can only be extended in its domain decomposition, then we should use the ideal decomposition that will naturally produce class group and class number.In a sense,the size of the class number can be used to measure how far apart the number of whole Dedekind ring and only integer ring.So far the study of the structure of the class group and the class number have a long history,the role of class group gets a good interpretation of class field theory and the role of class number is irreplaceable in number theory.Two frequency domains and sub-domains circle did not only get the attention of many mathematicians,but also have a lot of nice conclusions,though were not completely resolved.Some mathematicians got involved in the research about the class group and class number of cubic field,but the conclusion is not much.Cubic field is an important part of algebraic number theory,it permeates a lot of content in algebraic number theory.It’s very necessary to study the class group and class number of cubic field.This will promote the study of the other parts of cubic field.Minimal polynomial of cubic field have two types:one is f(x)=x3+b,the other is f(x)=x3+ax+b.The article uses the knowledge of algebraic number theory to study the class group and class number of totally real cubic field, K=Q(ω), in which ω is root of f(x)=x3-ax+b that is irreducible(which a=1～10,b=-10～1,a,b∈Z).The article has two chapters.Chapter1:introduction of the development process of algebraic number theory, the class group and class number’s background,outcomes,meaning and the basic knowledge the paper needs.Chapter2:apply knowledge and methods of algebraic number theory to study the class group and class number of totally real cubic field, K=Q(ω), in which co is root of f(x)=x3-ax+6that is irreducible(whicha=1～10,b=-1～-1,a,b∈Z).And get heir class group and class number.