MA/MSCMT-01
December 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.
Section A 8 × 2 16
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 derived subgroup.
Define Euclidean ring.
Define dual base.
Define normal extension of a field.
Define eigen vector.
Write Pythogoras theorem in inner product space.
Write Bessel's inequality.
(viii) Define orthogonal linear transformation.
068
MA/MSCMT-01 1800 3 (P.T.O.)
MA/MSCMT-01 1800 3 (Contd.)
068
Section B 4 × 8 32
Note: Section contain 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.
State and prove class equation of a group.
Let R be a Euclidean ring with a Euclidean valuation d. Then prove
that d is minimal among all d for non zero a ∈ R and u ∈ R
is a unit if and only if d d
If F C K C E are fields with and finite, then prove
that E/F is finite extension and

Prove that an n × n matrix A over a field F is invertible if and only
if rank n.
Let A [ai be any n × n matrix over a field then prove that
det Σ ∈ aσ 1 aσ 2 ............ aσ n
σ ∈ Sn
Prove that every orthonormal set of vector is a linearly independent
set in an inner product space.
Let t V → V1 be a linear transformation and V is finite
dimensional, then prove that
dim V rank nullity
If F is a field, then prove that every polynomial f ∈ F has
splitting field.
MA/MSCMT-01 1800 3
068
Section C 2 × 16 32
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. Use of
non-programmable scientific calculator is allowed in this
paper.
10) Prove that a group G is solvable if and only if for
some n ∈ N.
If K is a field and if σ1, σ2, ..........σn are distinct automorphisms
of then prove that it is impossible to find elements a1, a2,
..........an not all zero in K such that a1 a2 ............
an 0 for all u ∈ K.
11) Let R be a Euclidean ring. Then prove that any finitely generated
R module N is the direct sum of a finite number of cyclic sub modules.
12) Let t V → V be a linear transformation from a finite dimensional
vector space V to itself. Assume that i ....... n are
distinct eigenvectors of t corresponding to distinct eigenvalues
λi i ........ n. Then prove that ....... is a linearly
independent set.
If 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⊥.
13) State and prove principle axis theorem.