Primes P The Prime Ideal Decomposition Of Q (u ~ (6/1))

Since Kummer gave the definition of ideal and Dedekind developed the theory of ideal, Prime ideal decomposition is an important problem in algebraic number theory which has extensive function of the Diophantine Equation and the class field theory. Especially, it is strong to solve a pile of unsolved problems in the Diophantine Equation. Hence it is urgent and significant to factor the prime ideal of Q in its finite extension field.Q is the field of rational numbers, p is Q prime ideal, and R is the valuation ring. x~6 - u(u∈R) is a unique monica irreducible polynomial rational numbers Q . There are two ways to factor the prime ideal of Q in its finite extension field. One is extension and moving, the other is local-integral ideal. By local-integral ideal, this paper gets than whether a congruence equation has solutions, we investigate the decomposition of polynomial x~6 - uin a local field to determine the decomposition of a prime p in Q(?) . It has three cases:(1) (p,6) = 1,(p,u) = 1, (2) p|6,(p,u) = 1, (3) p~α|| u. And we established the possible type of decomposition of prime ideal.