Decomposition Of Prime Ideal（p） InQ(μ^1/35)

The decomposition of prime ideal is an important and typical topic in algebraic number theory. Then, the study of the finite extension decomposition of a prime ideal (p) in rational number Q has important significance. In this paper, we discuss the decomposition of a prime ideal (p) in Q((?))We first discuss the decomposition of the prime ideal in Q(ξ35), where is an extension of (p) in Q(ξ5). We know that prime ideal decomposition have a transitivity, as a result, We give the decomposition of in Q((?),ξ5), and obtain the decomposition of (p) in Q((?)) by the transitivity of prime ideal decomposition. From literatures we know that the decomposition of(p) in Q((?)) is determined by the decomposition of the prime ideal in Q((?),ξ35), where is an extension of (p) in Q(ξ35).We also discuss all possible decomposition of (p) in Q((?)), and determine the decomposition of (p) in Q((?),ξ35) and Q((?)) completely.