Iwasawa Main Conjecture Over Rational Number Field

This thesis is a review of the proof of Iwasawa’s conjecture in the field of rational numbers.This thesis is mainly based on the proof of Iwasawa’s conjecture in the whole real field by Andrew Wiles in 1990,and the conclusion of μ invariants by Ferrero and Washington in 1979.This subject covers a large number of new discoveries in mathematics in the 20th century,and Iwasawa,Ferrero,Washington,Mazur,Hida,and Wiles have done important work in this field.Iwasawa’s theory later played a crucial role in proving Fermat’s Last theorem and in the BSD conjecture.Iwazawa’s conjecture predicts that the analytic p-adic L function is equal to the algebraic p-adic L function,which on the one hand gives the representation of the determinant of algebraic objects and on the other hand means that the analytic objects imply profound information in the ideal group.In the first chapter,Iwasawa theory is introduced,and the exact definitions of analytic p-adic L function and algebraic p-adic zeta function are given respectively,and the description of conjecture is given.Through Weierstrass preparation theorem,the proof is divided into two parts:adjoint polynomial equality and μ invariants equality,and the proofs of these two parts are given in Chapter 2 and Chapter 3 respectively.In Chapter 2,we will prove the non-divergent extension of Iwasawa’s conjecture from the Λ-adic form of Hida,and give the equality of adjoint polynomials.The third chapter proves the disappearance of μ invariants by giving a counter example of Iwasawa’s discriminant.