Distributions Of Abelian Ramification Groups And Ideal Class Groups

In this thesis,we study the distributions of the abelian p-ramification groups on quadratic fields and of the 2-class groups on Kummer towers.In the first Chapter,we introduce the background and state our main results.In Chapter 2,we recall some preliminaries.Firstly,we review the class field theory and the ambiguous class number formula.In particular,we introduce the Gauss genus theory and the Redei matrix on the quadratic field.The second part presents the basic setting of the Cohen-Lenstra heuristic and introduce its application and progress on class groups.In Chapter 3,we study the abelian p-ramification theory of the quadratic field.For the imaginary quadratic fields,we prove an analog result of genus theory and Redei matrix.We also study a special class of 2-ramification groups on the real quadratic fields.Finally,we propose a Cohen-Lenstra conjecture on the abelian p-ramification groups.In Chapter 4,we use the Redei matrix and some analytical methods to prove the 4-rank density result of the abelian 2-ramification groups on the imaginary quadratic fields.In Chapter 5,we study the distributions of 2-class groups and fundamental units of the pure quartic fields Q((?)).In the last Chapter,we generalize the results of the quartic fields to the Kummer towers.In the appendix,we list the numerical data to support the conjectures on the abelian p-ramification groups.