The Decomposition Of Prime P In □(u～(1/n))

Prime ideal decomposition is an important problem in algebraic number theory and is relative closely with the class field theory and so on. Hence it is significant to factor the prime ideal of K in its finite extension field.By local-integral ideal, this paper gets that whether a congruence equation has solutions determines a decomposition of a prime ideal of K in its extension. In detail, the decomposition of polynomial in local field determines the forms of factoring the prime ideal of K in its extension field.This paper mainly consider the case that K is the national number field, and its extension is a rational number field by adding a n-th root of rational integer u , and gets some results according to different number n. There are three cases: (1) n is a prime; (2) n is a square or cube of a prime; (3) n is a product of two different primes. In three cases, we investigate the decomposition of a polynomial f ( x)=x~n -u in a local field to factor a prime ideal of K in its extension.