Decomposition Of Prime Ideal （p） In Q(u^1/33)

The decomposition of prime ideal is a very important question of the algebraicnumber theory. The question of decomposition of prime ideal （p）in the rationalGalois extension number field is a very significant problem.In this paper, the problem is divided into six steps. Firstly, according to thedecomposition forms of （p）over two cyclotomic fieldsQ（ζ₃）andQ(ζ₃₃),itgets the decomposition forms of A over the cyclotomic fieldQ (ζ₃₃), here A aexpanded prime ideal of （p）overQ（ζ₃）.Secondly, it proves the samedecomposition forms that both11times Galois extensionfields—Q(u^1/11, ζ₃₃) Q (ζ₃₃)andQ(u^1/11, ζ₁₁) Q (ζ₁₁)—have in threecircumstances, and gets the decomposition form of A overQ(u^1/11, ζ₃)；Thirdly,according to the article [1]（herelequals11）, it gets the specific decompositionforms of （p）overQ(u^1/11)；Fourthly, according to the specific results of the twoprevious parts, it gets the decomposition forms of T overQ（11u, ζ₃）, here T aexpanded prime ideal of （p）overQ(u^1/11). Fifthly, according to the fourth, itseparately gets the decomposition forms of T overQ(u^1/33)based on3timesGalois extension fields Q(u^1/33, ζ) Q(u^1/33, ζ₃); Finally, it puts the two specificresults of the third and fifth together, then solves the problem.In the first chapter, this paper briefly introduces the history and current situationof the study about prime ideal decomposition problem, simply mentioning theexisting research results at home and abroad; In the second chapter, it explains themeaning of used symbols and lists related concepts; In the third chapter, it introducesrelated theorems; In the last chapter, it gives the derived course and related results.