Research On Several Special Types Of The Quadratic And Cublic Fields With A Fundamental Unit

There are three chapters in this thesis, the author give some special funda-mental units of quadratic and cubic fields and conclusions.In chapter 1:Firstly introduce the research background and basic knowledge of the algebraic number theory to prepare for the following text; Secondly introduce the basic concepts and symbols to be used in this paper, and the main results of this paper.In chapter 2:Give the methods and steps of a fundamental unit in real quadratic field and use this step to determine if the Diophantine x2-dy2=N equation has integer solutions after researching the relationship between the unit group of quadratic field integer ring OK and real quadratic field Q((?)) Pell equation having integer solutions;at the same time, give the fundamental unit whenO((?)) has a fundamental unit of normal 1 in some special circumstances and use Dirichlet's to determine if the Diophantine t2-du2=2 equation has integer solutions.In chapter 3:Firstly give an integer base proof of a special cubic field on the base of researching the unit groups of quadratic field and the necessary and sufficient condition of having unit number in cubic field;Secondly propose two types of special cubic fields which have the fundamental unit of normal of 1.