課程名稱︰代數導論二
課程性質︰必修
課程教師︰陳榮凱
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰98/04/13
考試時限（分鐘）：150分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. (20 pts)
Let f(x)=x^4+1. Show that f(x) is irreducible in Z[x] and in Q[x].
2. (20 pts)
Let f(x)≠0 屬於 Q[x] be a polynomial and u 屬於 C such that f(u)=0.
Prove the following:
(a) The set I:={ g(x) 屬於 Q[x] | g(u)=0 } is an ideal of Q[x].
(b) There is an irrducible polynomial p(x) such that I=(p(x)).
3. (20 pts)
Give two examples of prime numbers p1,p2 屬於 Z such that (p1) is a prime
ideal inand (p2) is not a prime ideal in Z[i].
(Hint: You might need the fact that Z[i] is a Principle Ideal Domain.)
4. (10 pts)
Let R be a commutative ring with unity. Prove that an ideal I <| R is maximal
if and only if R/I is a field.
5. (15 pts)
Let D be an integral domain and S 包含於 D be a multiplicative set. Prove
that every ideal of S^(-1)R is of the form S^(-1)I:={ [a,s] | a屬於I,s屬於S}
for some I <| D.
6. (15 pts)
Let I be a prime ideal of Z[x] containing (p) and let D:=Z[x]/I be the
quotient ring. Show that the natural surjective homomorphism φ:Zp[x] → D
factors the natural map π:Z[x] → D. That is φ。ψ=π where ψ:Z[x] → Zp[x]
is defined naturally.
(Hint: You might need the fact that ideals I包含於J then there is a surjective
map R/I → R/J. Prove this, if you need this fact)
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