课程名称︰代数二 课程性质︰数学系选修，可抵必修代数导论二 课程教师︰林惠雯 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2017/5/22 考试时限（分钟）：50分钟 试题 : 1.(10 pts) True of false. Determine whether the following ten statements (a1)- (a5) and (b1)-(b5) concerning the ring of Gaussian integers Z[√(－1)] and the polynomial ring Z[x,y] of two variables x, y over Z are true. Write a "T" for each true statement, and write an "F" for each false statement. You don't have to explain your answers. (a1) Z[√(－1)] is a PID. │(b1) Z[x,y] is a PID. │ (a2) Z[√(－1)] is a UFD. │(b2) Z[x,y] is a UFD. │ (a3) Z[√(－1)] is a Noetherian ring.│(b3) Z[x,y] is a Noetherian ring. │ (a4) Z[√(－1)] is an Artinian ring. │(b4) Z[x,y] is an Artinian ring. │ (a5) Z[√(－1)] is a Dedekind domain.│(b5) Z[x,y] is a Dedekind domain. 2 2 3 2.(15 pts) Find a Grbner basis for the ideal 〈x ＋2xy , xy＋2y －1〉 of the polynomial ring C[x,y] with respect to the lexicgraphic ordering x＞y. Justify your answer. 3.(10 pts) Suppose R and S are integral domains such that R ⊆ S, and such that S/R is an integral extension. Prove that S is a field ⇔ R is a field. 4.(15 pts) Consider the following three affine algebraic sets cver C: 2 2 3 V := {(x,y) ∈ A ：y －x = 0 }; C 2 2 3 W := {(x,y) ∈ A ：2y －3x = 0 }; C 1 X := A C Prove that V and W are isomorphic as affine algebraic sets, but V and X are not isomorphic as affine algebraic sets. (Hint: You may consider their affine coordinate rings C[V] = C[x,y]/I(V) etc.) --

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