MA/MSCMT-01
December Examination 2017
M.A./M.Sc. (Previous) Mathematics Examination
Advanced Algebra
Paper MA/MSCMT-01
Time 3 Hours Max. Marks 80
Note: The question paper is divided into three sections B and C.
Section A 8 × 2 16
(Very Short Answer Type Questions)
Note: Section contains 08 Very Short Answer type questions. Examinees have to attempt all questions. Each question is of 02 marks and maximum word limit may be thirty words.
Define internal direct product of groups.
Write class equation of a group.
Define solvable sub group.
Define prime element in a ring.
Define module endomorphism.
Define simple field extension.
Define Galois extension of a field.
(viii) Define nullity of a matrix.
068
MA/MSCMT-01 2000 3 (P.T.O.)
068
MA/MSCMT-01 2000 3 (Contd.)
Section B 4 × 8 32
(Short Answer Type Questions)
Note: Section contains 08 Short Answer type questions. Examinees
will have to answer any four questions. Each question is of
08 marks. Examinees have to delimit each answer in maximum
200 words.
Prove that the set of all automorphism of G Aut is a group under
composition of automorphisms.
Prove that a group G is abelian if and only if where e is
identity of G and is derived subgroup of G.
If a group G has a solvable homomorphic image whose kernel is
solvable, them prove that the group is solvable.
Let F be a field and such that deg n 1.
Then prove that there exist a finite extension K of F in which
gets a full set of n roots such that 7K FA
For every prime p and n prove that these exists a finite field with
pn elements.
Prove that every orthonormal set of vectors is a linearly independent
set in an inner product space.
Let V be a finite dimensional inner product space and W be its any
subspace. Then prove that V is the direct sum of W and W=.
Prove that a polynomial of degree n over a field F can have at most
n roots in any extension field.
068
MA/MSCMT-01 2000 3
Section C 2 × 16 32
(Long Answer Type Questions)
Note: Section contain 04 Long Answer type questions. Examinees
will have to answer any two questions. Each question is of
16 marks. Examinees have to delimit each answer in maximum
500 words.
10) State and prove unique factorizaton theorem.
11) Let M be an R-module and N1 N2 ................ Nk be sub module
of m. Then prove that the following statements are equivalent:
M N1 5 N2 5 ........... 5 Nk
pf n1 n2 ......... nk 0 then
n1 n2 ...........= nk 0 for ni Ni
and
Ni (N1 N2 ..... Ni 1 Ni 1 ....... Nk)
12) Prove that any two finite dimensional vector space over the same
field are isomorphic if and only if they are of same dimension.
13) Let V and be finite dimensional vector space over a field F with
bases B and respectively. If t v if a linear transformation
then prove that
Where is the dual map of t and are the dual bases of B and
respectively.