課程名稱︰代數導論一
課程性質︰數學系大二必修
課程教師︰李秋坤教授
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2020/1/8
考試時限（分鐘）：130
試題 :
(總分160分)
(試題中的屬於符號用ε代替, isomorphic to 的符號用～代替)
Throughout, a ring or an algebra always contains a unity 1.
1. Multiple Select Multiple-Choice Questions. For each of the following
questions, select one or more correct answers. Do NOT verify them.
(i) (12%) Let G be a group and N be a normal subgroup of G. Which of the
following statements always imply N is a normal subgroup of G?
(A) gNg^-1=N for all gεG
(B) gNg^-1 is included in (包含於) N for all gεG
(C) N is included in (包含於) gNg^-1 for all gεG
(ii) (8%) Let R be a ring and I be an additive subgroup of R. Which of
the following statements always imply I is an ideal of R?
(A) I=aR for some aεR
(B) I=aR for all aεR
(C) I=Ra for some aεR
(D) I=Ra for all aεR
2.
(i) (10%) State the First Isomorphism Theorem for groups. Don't prove it.
(ii) (10%) Let G1 and G2 be groups and G=G1xG2 be the direct product of
G1 and G2. Let N={(x,e2)εG | xεG1}, where e2 is the identity
element of G2. Show that N is a normal subgroup of G and G/N～G2
3. (i) (10%) State the Lagrange's theorem. Don't prove it.
(ii) (10%) Let Q be the quaternion group. Find all left cosets of H={1,-1}
in Q.
4. (20%) Let R be a ring and I be an additive subgroup of R. Prove that the
following statements are equivalent.
(i) I is an ideal of R.
(ii) For a, a', b, b'εR, a+I=a'+I and b+I=b'+I imply ab+I=a'b'+I.
5. (20%) Let n be a positive integer. Show that nZ is a maximal ideal of Z
if and only if n is a prime.
6. (20%) Let f(X), g(X)εR[X] be non-constant polynomials. Suppose that
f(X)+1/f(X)=g(X)+1/g(X) holds in R(X). Prove that f(X)=g(X).
(此處的R[X]代表polynomial ring over real numbers,
R(X) 則代表rational functions over real numbers)
7. (20%) Let I and J be ideals if a commutative ring R. Prove that I+J=R
implies IJ=I∩J.
8. Let A be a finite-dimensional real algebra of dimension n＞0.
(i) (10%) Let s be an invertible element of A. Prove that sts^-1≠t+1
for all tεA.
(ii) (10%) Letφ be an algebra automorphism of A. Prove that φ(t)≠t+1
for all tεA.