Gauhati University 2007 Previous : thetics: -III - Question Paper

2007
MATHEMATICS
THIRD PAPER
(Algebra)
Full Marks: 80
Time: three hours
The figures in the margin indicate full marks for the ques.
PART-A (Objective-type Questions)
(Marks: 32)
1. Fill in the blanks: 4
(a) The cyclic subgroup of Z24 generated by 18/ has order_________________.
(b) Z3 × Z4 is of order_____________________.
(c) The element (4, 2) of Z12 × Z8 has order _________________.
(d) The Klein 4-group is isomorphic to___________________.
2. Classify the group
Z4 × Z6 / (2, 3)
Where (2, 3) ?Z4 × Z6 and (2, 3) is the subgroup of Z4 × Z6 generated by (2, 3). 4
3. provide an example of a subnormal series which is not a normal series. 4
4. Justify the statement “Z has no composition series”. [Z is the additive group of
integers.] 4
5. obtain the value of
(i) (x+1)2
(ii) (x+1)+(x+1)
in Z2[x]. 2+2
6. Write the ideals of Z. Write the maximal ideal of Z. 4
7. The polynomial p(x) = x2+x+1 in Z2[x] is irreducible over Z2. Justify 4
8. Mark whether the subsequent statement is actual or not and justify your answer: 4
Q is an extension field of Z2 (with usual notations).
PART-B (Subjective-type Questions)
(Marks: 48)
ans any 3 ques.
9. (a) If G is a group and H, K are subgroups of G such that
G = H × K
Prove that
H  G/K and K G/H 6
b) Let H be a normal subgroup of a group G. If both H and G/H are solvable, then
show that G is solvable. 6
c) If 2 groups H and K are solvable, then show that H × K is also solvable. 4
10. (a) describe Euclidean domain with 1 example. Show that every field is a Euclidean
domain. 2+4
(b) In Z / (6), show that two is a prime element but not irreducible. 6
(c) State Eisenstein’s irreducibility criterion for a polynomial over the field Q. Use it
to show that the subsequent polynomials are irreducible over Q: 2+2
(i) x4- 4x +2
(ii) x3 - 9x+15
11. (a) If L is a finite extension of a field K and K is a finite extension of a field F, then
show that L is a finite extension of F. 6
(b) If a field F has q elements, then show that F is a splitting field of xq - x over its
prime subfield. 6
(c) Show that is algebraic over Q. 4
12. (a) For any 2 subspaces W1 and W2 of a vector space VF, show that W1+W2 is the
subspace of V spanned by W1 W2. 6
(b) Let S and T be the linear operators on R2 described by S(x, y) = (0, x) and T(x, y) =
(x, 0). Show that
(i) TS = 0
(ii) ST 0
(iii)T2 = T
(c) obtain the matrix representation of the subsequent linear operator T on R3 relative to
the usual basis {e1= (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1)}:
T: R3  R3 by
T(x, y, z) = (2x-3y+4z, 5x-y+2z, 4x+7y)
13. (a) Suppose A and B are similar matrices. Show that A and B have the identical
characteristics polynomial. 4
(b) Let
Is A similar to a diagonal matrix? If so, obtain 1 such matrix. 6
(c) Determine all possible Jordan canonical forms for a linear operator T: VV whose
characteristic polynomial is 6
2 + three 5






  


=
0 0 3
0 two 3
1 two 3
A
( ) ( )3 ( )2 D t = t - two t - 5

Earning:   Approval pending.