The Decomposition Of Prime Number P In The Domain Q(u^1/10)

The decomposition of prime ideal is very important thesis in the algebraic number theory and is relative closely with the class field theory and so on. Kummer firstly solved the decomposition of a prime p in ring of algebraic integers O_K. Gauss investigated the solvability of the quadratic equation x²-d≡0(modp)which combined with the well-known law of quadratic reciprocity and got the decomposition characteristics of prime numbers in number fields.Prime ideal decomposition has two ways: expansion of translation and local fields.By local-integral ideal, this paper gets that whether a congruence equation has solutions. We investigate the decomposition in a local field to determine the decomposition of a prime p in Q(u^1/10).Moreover we established the possible type of decomposition of prime ideal. Main idea is that the decomposition of a prime p in Q(u^1/10) is transformed intoQ(u^1/5)and Q(u^1/2).In the first part,we give the summarize for the condition and the significance. In the second part, we give the preparation knowledge of the whole paper and reveal in detail the source of the decomposition of prime ideal. In the third part, we investigate the decomposition in a local field to determine the decomposition of a prime p in Q(u^1/10),which has three cases (1) ( p |10,(p,u)=1),(2) (( p ,10) = 1,( p , u) = 1),(3)( pα|| u).