Residue Fields Of Gauss Extensions On The Quotient Field Of KZ^{(n)}

Skew group rings are very important rings in algebra.Graded extensions over skew group rings are a class of ring extensions with good properties.According to previous s-tudies,there is a one-to-one correspondence between the set of graded extensions and the set of Gauss extensions.So we can study the algebraic structure of Gauss extensions by studying the algebraic structure of graded extensions.Skew Laurent polynomial ring is an extremely significant ring.According to the properties of A1 and A₁,Xie Guangming and H.Marubayashi divided the graded extensions of K[Z,σ]into type（a）,type（b）,type（c）,type（d）,type（e）,type（f）,type（g）,type（h）.On the basis of the former studies,Li Haihe discussed the graded extensions of V in KZ（n）in details.Let K be a,field and Vbe a total valuation ring on K.A=（?）u∈Z（n）AuXu is a graded extension of V in KZ（n）and R=AJg（A）is the localization of the graded Jacobson radical of A.As can be seen from literature[17],R is a Gauss extension,（?）＝R/J（R）is isomorphic to the fractional field of （?）=A/Jg（A）.Therefore,we can study the related algebraic structure of the residue fields of Gauss extensions by studying the structure of （?）.This paper contains the follow five parts.The first part:we introduce the research backgrounds.The second part:Chapter 2 is based on Chapter 1.According to type（a）,type（d）,type（e）and the generalized type（h）and their special type（g）,we study the the correspond-ing structures and properties of residue fields of Gauss extensions on quotient field of KZ（n）The main lemma is 2.5;the main theorem is 2.6.Lemma 2.5 Let A=（?）u∈Z（n）AuXu be a graded extention of V on KZ（n）,H={u∈Z（n）|AuA-u =V}.Suppose that r（H）=r>0,H=L（u1,u2,...,ur）,and Aui= Vci（i=1,...,r）.Let Yi =（?）＝CiXui+Jg（A）.Then Y1,...,Yr are independent indeterminates of （?）.Theorem 2.6 Let A=（?）u∈Z（n）AuXu be a graded extention of V on KZ（n）,H is defined as the lemma above.Let R=AJg（A）,R=R/J（R）.Then （?）[Y1,...,Yr],（?）（Y1,...,Yr,）.Y1,...,Yr are independent indeterminates of （?）.Chapter 3 gives some lemmas and related examples of residue fields of Gauss extensions when V is a discrete valuation ring.The main theorem is 3.5.Theorem 3.5 Let V be a discrete valuation ring of K.A=（?）u∈Z（n）Au,Xu is a graded extention of V on KZ（n）.Then A must be graded extensions of type（a）or type（e）.（1）If A is a graded extension of type（a）,then H=Z（n）.Let R=AJg（A）,（?）=R/J（R）.Then （?）[Y1,...,Yn],（?）（Y1,...,Yn）.Y1,...,Yr are independent indeterminates of （?）.（2）If A is a graded extension of type（e）,f is a nonsingular graded mapping corresponding to A.Let H = {u∈ z（n）| f（u）+f（-u）= 0}.Suppose that R（H）= r.Let R =AJg（A）,（?）= R/J（R）,Then （?）[Y1,...,Yr],（?）（Y1,...,Yr）.Y1,...,Yr are independent indeterminates of （?）.In Chapter 4,we illustrate the algebraic structures of different classes of residue fields of Gauss extensions.The last part is the conclusion.We summarize the research works of this paper,and put forward some expanded problems.