The Decomposition Of Prime Ideal(p)in The Domain Q(85√u)

As one of the most important issues in the algebraic number theory, the decomposition of prime ideal is always a concern.How to determine the decomposition forms of prime ideal of the basic field in finite extension is undoubtedly important.According to the existing conclusions of the decomposition of prime ideal,we study the problem from various perspectives.First of all,we make a detailed study of the problem of the decomposition of prime ideal A which is the prime ideal in the rational number field extension Q(ξ5) in Q(ξ85) and Q((?),ξ5).Then we obtain the decomposition of prime ideal (p) in the rational number field extension Q((?)) from literature [4].Secondly, we obtain the decomposition of prime ideal T which is the prime ideal in the rational number field extension Q((?)) in Q((?),ξ5) and Q((?)). Finally, we discuss all of the decomposition conditions and forms of (p) in the rational number field extension Q((?)).Then the question of decomposition of prime ideal (p) of the rational number field in the rational number field extension Q((?)) has been solved perfectly and completely.In the first chapter,we give the summarize for the development history and the existing results.In the second and third chapter, we give the related definitions and theoretical basis.The chapter four,we give a detailed proof by the basic knowledge and lemma that the first three chapters were given.