The Decomposition Of The Prime Number P In The Domain Q(u^1/18)and Q(u^1/lm)

Since Kummer gave the definition of ideal, and Dedekind developed and formed the theory of ideal, the decomposition about prime ideal has been paid more and more attention. How to determine the decomposition situations of the prime ideal in limited expansion has become an important issue in the algebraic number theory. This paper discussed the problems whether the congruence equations have the solution, and get the value in the decomposition type ewhich the ramification index ei ,the residue class degree fi, the split times g in local field, and the specific circumstances of prime ideal decomposition are given. The main parts are three, four, five chapters in this article.In chapter 3, under the condition of p≠l, (p,u)=1, we investigate the problems how the prime number p take the value, the equation of x~ 9≡u(modp) and x~3≡u(modp) are solvable or not, then get the specific form of the prime p decomposition in Q (9 u).In chapter 4, combining Hensel lemma,Galois expansion, the local field method and some other theories. Under the condition of p≠l, (p,u)=1, we get the specific form of the prime number p decomposition in Q (18 u). And according to the conclusion about lemma 2.14, the paper is divided into two aspects in this chapter.In chapter 5, we adopt Eisenstein method, local field idea and consider the following three situations1) p≠l, (p,u)=1; 2) pa || u,(l,a)= 1,a∈N; 3) p = l, (p,u)=1,Then get the corresponding results of the prime number p decomposition in the field Q(?).