On The Galois Module Structure Of Ideal Class Groups

An algebraic number field is a finite field extension of Q.The ideal class group Cl(K)and class number hK of algebraic number field K reveals the difference between the law of the number in the integer ring OK of K and the law of the number in Z.Therefore,the study of the structure of Cl(K)and hK has always been one of the most important problems in algebraic number theory.For a general algebraic number field,there is no general way to analyze the characteristics of its ideal class group.Mathematicians began to study from the cyclotomic fields Q(?l)is a prime number),and this study was extended to more general fields.Up to now,there are many conclusions about the class groups of the cyclotomic fields and the quadratic fields.Frohlich was the first people who considered Cl(K)as Gal(K/Q)-module and applied the knowledge of group theory and representation theory to study its ideal class group when K is a normal extension on Q.Subsequently,this approach has been adopted by others.They derived many important conclusions,especially on the p-rank rp(Cl(K)):=dimFp(Cl(K)[P]of Cl(K),which gave us a better understanding of the ideal class groups.On the basis of these studies,this thesis proves that many conclusions about rp(Cl(K))(cf.[9],[10]and[14])can be extended to pn-rank rpn(Cl(K))of Cl(K)with the help of the method used in the discussion of rp(Cl(K)in[10],in which rpn(Cl(K)):=dimFp(Cl(K)[pn]/Cl(K)[pn-1])and n is an arbitrary positive integer.