Decomposition Of Prime Ideal （P） In Q(u^1/51)

Prime ideal decomposition problem algebraic number theory is an importantsubject, it and kind of domain theory is very close relationship, since the library’svision are ideal definition, and the ideal theory by development, the problem of thebreakdown of prime ideal in diophantine equation, the class domain theory has a wideuse in, especially for solving diophantine equation of many have not met in theproblem has a strong role, so how to judge element in domain of ideal situation islimited expansion of decomposition urgent and the value.Let Q to rational number domain, p for Q prime ideal, R for the assignmentring, x51uthe rational number in the domain is irreducible polynomials. Theproblem of the breakdown of ideal, basically have two kinds of methods, one kind isthe translation of the expansion of the method; Another kind is to use local domainmethod. This article will use the two methods to discuss prime ideal p in therational number domain ideal Q51times root expansion of decomposition Q (u^1/51),at the same time, gives all the decomposition forms of prime ideal p in Q ((u^1/51).