Padmashree Dr DY Patil Vidyapeeth 2007 M.Sc Mathematics Mathamtics -university question paper

M.Sc. DEGREE EXAMINATION, 2007
( MATHEMATICS )
( 1st YEAR )
( PAPER - I )
110. ALGEBRA

(Revised Regulations)
May ] [ Time : three Hours
Maximum : 100 Marks
PART ? A (8 × five = 40)

ans any 8 ques..

every ques. carries 5 marks.
1. Let H, K be 2 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 Î V are linearly independent,
prove that every victor in their linear span
has a unique representation in the form
l1v1 + l2v2 + ....... + lnvn with the lnÎ 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 Î A(V) is singular if and
only if here exists a non ? zero vector
v in V such that (v)T = 0.

9. If ÎA(V) is Hermitian, prove that all
its characteristic roots are real.

10. If is unitary and if l is a characteristic
root of , prove that | l | = 1.

PART ? B (3 × 20 = 60)

ans any 3 ques..

every ques. carries TWENTY marks.

11. (a) Let G, be groups Let f be a homomorphism
of G onto with Kernel K. Prove that is
isomorphic o .

(b) If p is a prime number and if pa divides
O(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
maximal ideal of R if and only if is a
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 2 bases of V have
the identical number of elements.

(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 Î A(v) and if S is regular, prove that
T and S T S?1 have the identical 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 2 finite fields having the identical
number of elements are isomorphic.

Earning:   Approval pending.