Gauhati University MathematicsAlgebra 2007 Question Paper

University :  Gauhati University

Degree : B.Sc Mathematics 

Question Paper Subject: Algebra

Question Paper Year : 2007

Question Paper Details : Previous Years Question Papers Types Version

MATHEMATICS

THIRD PAPER

(Algebra)

The figures in the margin indicate full marks for the questions

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. Give 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. Find 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 following statement is true or not and justify your answer: 4

Q is an extension field of Z2 (with usual notations).

PART-B (Subjective-type Questions)

(Marks: 48)

Answer any three questions

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 two groups H and K are solvable, then show that H × K is also solvable. 4

10. (a) Define Euclidean domain with one example. Show that every field is a Euclidean

domain. 2+4

(b) In Z / (6), show that 2 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 following 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 two 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 defined by S(x, y) = (0, x) and T(x, y) =

(x, 0). Show that

(i) TS = 0

(ii) ST 0

(iii)T2 = T

(c) Find the matrix representation of the following 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 same

characteristics polynomial. 4

(b) Let

Is A similar to a diagonal matrix? If so, find one such matrix. 6

(c) Determine all possible Jordan canonical forms for a linear operator T: V V whose

characteristic polynomial is 6

2 + 3 5

=

0 0 3

0 2 3

1 2 3

A

( ) ( )3 ( )2 D t = t - 2 t - 5 

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