Decomposition Of Prime Ideal (p) In Q(u～(1/11))

The decomposition of prime ideal is not only very important thesis in the algebraic number theory, but also plays a key role in the way that all Galois extension solved on the basic field F . So many researchers study to the extensive and higher the decomposition of prime ideal in the internal and the external.Let Q be the field of rational numbers, and (?) be its valuation of rank 1 non-trivial and non-Archimedean, R is the valuation ring with respect to (?), p is the prime ideal with respect to R. In this paper, x¹¹ -u(u ∈ R) is a unique monic irreducible polynomial rational numbers Q. if K/Q is rank 11 Galois extension, we have discussed the question of decomposition of prime ideal (p) in the rational number fieldextension Q(u^1/11) and solved completely.The first part, we give the summarize for the condition and the significance;the second part, we give the preparation knowledge the whole paper and reveal in detail the source of the decomposition of prime ideal. The third part, we give a whole proof by the knowledge and proposition that the second part was given.