課程名稱︰代數二
課程性質︰數學系選修，可抵必修代數導論二
課程教師︰林惠雯
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/5/22
考試時限（分鐘）：50分鐘
試題 :
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.)