Vinayaka Missions University 2008 M.Sc Mathematics ALGEBRA - 1 : t - Question Paper

COURSE CODE - 2030501
PG DEGREE exam – SEP 2008
M.SC (MATHS)
ALGEBRA - 1
(For Candidate Admitted from calendar 2007 onwards)
Time: three Hours Max. Marks: 75
part –A
ans all the Questions: 15X1= 15
1. Example for a Group?
2. describe sub – Group?
3. Condition for homomorphism?
4. Express as the product of disjoint cycles (1,2) (2,3) (1,2)?
5. If p is a prime number and p/o (G),then__________
6. Real numbers under addition and multiplication is a ring or
not_______________________
7. Every Commutative Ring is an______________
8. A finite Integral domain is a ________________
9. describe Ideal?
10. If R is a unique factorization domain and if f1 g E R then c (f g) =
________________________________
11. If f (x) and g(x) are primitive polynomials then f (x)
g(x) is a
12. If F is a field then F(x1,x2………..x n) is a
13. describe a vector space homomorphism?
14. In a vector space ?(v-w)=
15. describe a linear Consbination?
part – B
ans any 5 questions:- five X6=30
16. a)Define and Example for Group, Normal Sub Groups & Quotient
Groups?
(Or)
b) Let f be a homomorphism of G onto G with kernel K
and let N be a normal subgroup of G, N={XEG / f (x) E N }.then
G/N ˜G / N. Equivalently, G/N ˜ (G/K)/(N/K).
17. a)Every permutaion is a Product of its cycles?
(Or)
b)If 0(G)= p2 where P is a prime number then G is abelian?
18. a)If f is a homomorphism of R into R1 with kernel I (f) , then
a. I (f) is a subgroup of R under addition?
b. If a E I f AND r E R then both ar &ra are in I (f)
(Or)
b)If R is a commutative ring with unit element and M is an ideal of
R, then M is a maximal ideal of R iff R/M is a field?
19. a)State and prove ”The Eisenstein criferion”?
(Or)
b)State and prove “unique Factorication theorem”?
20. a)L (S) is a subspace of V?
(Or)
b)Any finite abelian group is the direct product of cyclic groups?
part – C
ans any 2 Questions:- two X15=30
21. If G is a group, then A(G), the set of automorphisms of G,is also a
group?
22. State and prove cayley’s theorm?
23. The ideal A=(a0) is a marimal ideal of the Euclidean ring R iff a0 is
a prime element of R?
24. Let R be a commutative ring with unit element whose only ideals
are (o) and R itself. Then R is a field?
25. If V is finite – dimensional and if W is a subspace of v1 then W is
finite dimensional, dim W = dim V and dim V/W = dim V- dim
W.

Earning:   Approval pending.