課程名稱︰代數導論二
課程性質︰數學系必修
課程教師︰莊武諺
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰106/4/21
考試時限（分鐘）：120分鐘
試題 :
1.(15 points) Recall that in a PID R, the ideal I = (r)⊂R generated by
r ∈R is maximal if and only if r is irreducible in R.
Show that C[x,y] is not a PID.
~ ×
2.(15 points) Let R be an integral domain and R = R ∪{0} the collection
~
of units of R together with 0. An element u ∈R - R is called a universal
~
side divisor if for every x ∈R there is some z ∈R such that we have
x = qu + z for some q ∈R.
Show that if R is a Euclidean domain that is not a field, then there exists
a universal side divisor in R.
3
3.(15 points) Let θbe a root of the polynomial x -3x-3.
1 + θ 2
Please express