Tamil Nadu Open University (TNOU) 2009-1st Year M.Sc Mathematics Tamilnadu open university Maths Algebra - exam paper

M.Sc. DEGREE EXAMINATION JUNE 2009.

(AY 2005-2006 and CY 2006 batches only)

First Year

Mathematics

ALGEBRA

Time : 3 hours Maximum marks : 75

PART A (5 5 = 25 marks)

Answer any FIVE questions.

1.         Show that N of a group G is a normal subgroup of G if and only if every left coset of N in G in a right coset of N in G.

2.         Show that every permutation is the product of its disjoint cycles.

3.         Let R be a commutative ring with unit element whose only ideals are (o) and R itself. Show that R is field.

4.         Let R be an Euclidean ring and is not a unit in R show that d (a) < d (a,b).

5.         Let V be a vector space over a field F and let
S,. Show that L (SUT) = L(S) + L (T).

6.         Show that is algebraic over Q. Find its degree.

7.         Find the splitting field of the polynomial over Q.

8.         If K is a field of complex numbers and F is a field of real number compute G (K : F).

PART B (5 10 = 50 marks)

Answer any FIVE questions.

9.         State and prove fundamental theorem of group homomorphism.

10.       If p is a prime number and show that G has a subgroup of order .

11.       Show that every integral domain can be imbedded in a field.

12.       If F is a field show that F (x) is an Euclidean ring.

13.       (a) If V is a finite dimensional vector space and W is a subspace of V, show that .

             (b) If V is finite dimensional show that V is isomorphic to .

14.       Show that a field is algebraic over F if and only if F (a) is finite extension of F.

15.       If p (x) is irreducible in F (x) and if v is a root of
p (x) show that F (v) is isomorphic to F' (w) where w is a root of p' (+) and the isomorphism be so choosen such that

16.       If K is a normal extension of F and H is a subgroup of G (K : F). Let . Show that

             (a)

             (b) .