The Integral Bases Of Cubic Field And Cyclotomic Field

In this thesis,we discuss the integral bases in the two number fields,cubic field and cyclotomic field,and using the method of algebraic number theory solve the problem of the Diophantine equations.In chapterl,we introduce the development process of algebraic number theory. Proposing of Fermat's laid the foundation of the development of algebraic number theory. The proposed of solving the Fermat's theory is the proposing of establishment and development of algebraic number theory.Gauss,Kummer,Dedekind,Hualuogen and Fengkeqin has made tremendous contributions to the development of Number Theory.The problem of bases and unit in quadratic field has been solved, but in cubic field, the forms of bases we don't know. Even in cycle field, we don't know the forms of bases. The forms of bases of pure cubic field have been given by Blair K.Spearman and Kenneth S.Williams. In chapter 2,I introduce a easy method to find the basis of the pure cubic field. Further,we can know the discriminant of Algebraic number field.By the bases and unit in cubic field,we can solve the problem of Diophantine, this is a complicated problem, but we must know it. Although with the method of recurrence sequences and guadratic remainder, we can solve the Diophantine equation, in the field we can solve more Diophantine equation. In chapter 3,author proved the Diophantine equation x3-6x-3= 3y2 has no integer solution.Many people are interesting in power integral basis,normal integral basis, and relative integral basis, we know quadratic field and cyclotomic field have power integral basis. In chapter 4,we introduce the generators of power integral basis in cyclotomic field Q(ζ21),and find the generator of power integral of power integral basis in Q(ζ21).