課程名稱︰代數導論
課程性質︰系定必修
課程教師︰李白飛 教授
開課系所︰數學系
考試時間︰2006/01/09 10:00～12:50
試題： Algebra I
1. Show that a group of order 1975 is abelian. (10pts)
2. Let G be a simple group of order 168. How many elements of order
7 are there in G? (10pts)
3. Let R be a ring and n＞1 an interger. Show that
(a) M_n(R) is not a domain if R≠0. (10pts)
2
(b) M_n(R) is not commutative if R ≠0. (10pts)
4. Show that a finite domain is a division ring. (10pts)
n
5. Let R be a commutative ring with unity and f(x)＝a_0＋a_1‧x＋...＋a_n‧x
屬於 R[x]. Show that f(x) is invertible in R[x] if a_0 is invertible in R
and a_1,...,a_n are all nilpotent. (10pts)
6. Let A,B,C be subgroups of an (additive) abelian group with A＋B＝B＋C and
A∩B＝B∩C. Show that A＝B if A 包含等於 B or B 包含等於 A. Give an
example to show that A＝B need not be true in general. (10pts)
7. Show that two Gaussian intergers a,b are relatively prime in Ｚ[i] if δ(a)
and δ(b) are relatively prime in Ｚ. (10pts) Give two Gaussian intergers
a,b which are relatively prime in Ｚ[i] while δ(a) and δ(b) are not
relatively prime in Ｚ. (10pts)
2 2
8. Show that the equation 7x －15y ＝43 is not solvable in Ｚ. (10pts)
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※ 編輯: monotones 來自: 140.112.250.148 (01/18 11:40)