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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 -------------- as a Q-linear combination of {1 ,θ ,θ}
2
3 + 2θ+ θ
in Q(θ).
4.(15 points) Let K be a extension of the field F. Prove that K is a finite
extension over F if and only if K is generated by a finite number of
algebraic elements over F.
5.(15 points) Let R be a UFD with quotient field F and let p(x)∈R[x].
Prove that if p(x)=A(x)B(x) for some nonconstant polynomials A(x),B(x)∈F[x],
then there exist r,s ∈F such that rA(x)=a(x) and sB(x)=b(x) both are in R[x]
and p(x)=a(x)b(x) is a factorization in R[x].
6.(15 points) Show that
10 9 8 7 6 5 4 3 2
(i)x + x + x + x + x + x + x + x + x + 12x + 1 is irreducible in Q[x],and
n
(ii)X - t is irreducible in (C(t))[X], where n is an integer larger than 1.
(You don't have to prove the irreducibility criterion you are applying, but
you need to state the criterion explicitly in order to obtain full points.)
3
7.(15 points) Show that x - √2 is irreducible in Q(√2)[x].
8.(15 points) Show that Q(√2,√3) = Q(√2+√3).
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※ 編輯: ntumath (140.112.253.33), 04/28/2017 12:42:26