Anna University Chennai 2007 M.Sc Computer Science Algebra - exam paper
Algebra
4
(b) Let V be a finite dimensional inner product space. Prove that V has an orthonormal set as a basis.
14. (a) Prove that a polynomial of degree n
over a field can have at most n roots in any extension field.
(b) If T, S A(v) and if S is regular, prove that T and STS1 have the same minimal polynominal.
15. (a) If N is normal and AN = NA, prove that
AN* = N* A, where A is any linear transformation on V.
(b) Prove that any two finite fields having the same number of elements are isomorphic.
Name of the Candidate :
7 7 0 0 M.Sc. DEGREE EXAMINATION, 2007
( MATHEMATICS )
( FIRST YEAR )
( PAPER - I )
110. ALGEBRA
(Revised Regulations)
May ] [ Time : 3 Hours
Maximum : 100 Marks
PART-A (8x5 = 40)
Answer any EIGHT questions.
Each questions carries FIVE marks.
1. Let H, K be two subgroups of a group G. Prove that H K is a subgroup of G if and only if, H K = K H.
2. Let G be a group. Let A(G) be the set of all automorphisms of G. Prove that A(G) is also a group.
3. Prove that a finite integral domain is a field.
4. Prove that a Euclidean ring possesses a unit element.
5. If v1? v2 , .... vn e V are linearly independent, prove that every victor in their linear span has a unique representation in the form
1v1 + A,2v2 +.......+ nvn with the ne f.
6. State and prove Schwarz inequality.
7. Let F, K, L be fields. If L is an algebraic extension of K and if K is an algebraic extension of F, prove that L is an algebraic extension of F.
8. Let V be a finite dimensional vector space over a field F. Prove that If 1 e A(V) is singular if and only if A here exists a non - zero vector v in V such that (v)T = 0.
9. If leA(V) is Hermitian, prove that all its characteristic roots are real.
10. If 1 is unitary and if is a characteristic root of 1, prove that | X | = 1.
Answer any THREE questions.
Each questions carries TWENTY marks.
11. (a) Let G, G be groups Let c|) be a homomorphism
G
of G onto G with Kernel K. Prove that is
K
isomorphic -fc G.
oc
(b) If p is a prime number and if p divides 0(G), prove that G has a subgroup of order Pa.
12. (a) Let R be a commutative ring with unit element.
Let M be an ideal of R. Prove that M is a
R
maximal ideal of R if and only if is a
M
field.
(b) If f(x) and g(x) are primitive polynomials, prove that f(x) g(x) is a primitive polynomial.
13. (a) If V is a finite dimensional vector space over
a field F, prove that any two bases of V have the same number of elements.
Turn over
|
Attachment: |
| Earning: Approval pending. |
