MA/MSCMT-01
June Examination 2016
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. Write answer as per the given instruction. Use of non-programmable scientific calculator is allowed in this paper.
Section A 8 × 2 16
(Very Short Answer Type Questions)
Note: Section contain 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.
Define solvable group.
Define quotient module.
Define prime elements.
Define solvability by radicals.
Define ergen value.
Write Schwartz's inequality.
(viii) Define adjoint of a linear map.
Section B 4 × 8 32
(Short Answer Type Questions)
Note: Section contain 08 Short Answer Type Questions. Examinees
have to answer any four questions. Each question is of 08
marks. Examinees have to delimit each answer in maximum
200 words.
Prove that a group G is abelian if and only if G e is identify
of G.
Let R be a Euclidean ring and a and b be any two non zero elements
in R. If b is not a unit in then prove that d d
If f M u is an R-module homomorphism. Prove that f is a
monomorphism if and only if ker f
Prove that an irreducible polynomial f over a field F of
characteristics p 0 is inseparable if and only if f is a polynomial
in x p.
If v is a finite dimensional vector space over a field Prove that for
every non-zero vector v e V there exists a linear functional f in
s.t. f 0.
Let u be vector spaces over a field F. Let vj i w
1 " be bases for w and u respectively. If t V u s W u U
are linear transformations, A and B are the matrices associated with
t and s respectively. Then prove that the matrix associated with
is BA.
If a square matrix A of order over a field f has n distinct ergen
values m1, m2, ....., mn . Prove that there is an invertible matrix P such
that p-1 AP drag m2, ....., mn) .
Let v and v be inner product spaces. Prove that a linear transformation
t v u v is orthogonal if and only if t u for all u e v.
Section C 2 × 16 32
(Long Answer 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) Prove that every group is isomorphic to a group of permutations.
Let G1 and G2 be groups. If Ni is normal in Gi; i 2. Prove that
N1 × N2 is normal in G1 × G2 and (G1 × G2) (N1 × N2) ≅
(G1 × N1) × (G2 × N2)
11) Let k be the finite algebraic extension of a field F. Prove that k
is a normal extension of F if and only if k is the splitting field of
some polynomial over F.
If k is finite, separable and normal extension of a field F. Prove
that an element of k which remain invariant for each member of
Galo is group is necessarity a member of F.
12) Let V and V be finite dimensional vector spaces over a field F with
bases B and B respectively. If t V u V is a linear transformation
then prove that M M
l 6 where is the dual map of
and B are the dual bases of B and B respectively.
13) Prove that every finite dimensional vector space V with an inner
product has an orthonormal basis.