Decomposition Of Prime Ideal (p) In Q(65√μ)

Since Kummer built the definition of ideal and related theory, and then the theory of ideal was proved and developt by Dedekind, the problem of decomposition of prime ideal is an important topic in algebraic number theory. Furthermore, it is especially important to study lots of the issue of decomposition of prime ideal in the internal and external.In the paper, on the basis of fully draw on the existing theory of decomposition of prime ideal, adopting the method of expansion of translation and the method of the local domain, the decomposition of prime ideal (p) in the domain Q(65√μ) has been discussed.Firstly, in the paper, let A is the expansion prime ideal of prime ideal (p) in the rational field extension Q(ζ5).We make a detailed study of the question of decomposition of prime ideal A in the rational field extensionQ(65√μ)andQ(13√μ,ζ5) completely.By decomposition of prime ideal of transitivity property, we obtain decomposition of prime ideal (p) in the rational field extension Q(13√μ).Secondly, let P is the expansion prime ideal of prime ideal (p) in the rational field extension g(ζ65).Based on the reference literature,we prove the decomposition of prime ideal (p) in Q(65√μ) to be decided by the decomposition of prime ideal P in Q(65√μ,ζ65).Finally, we discuss all of the decomposition styles of (p) in Q(13√μ),and determine the decomposition of prime ideal (p) in Q(65√μ)and Q(65√μ,ζ65) completely.