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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