Tamil Nadu Open University (TNOU) 2009-1st Year M.Sc Mathematics Tamilnadu open university Maths Algebra - Question Paper
M.Sc. DEGREE exam JUNE 2009.
First Year
(AY 2006–07 batches onwards)
Mathematics
ALGEBRA
Time : three hours Maximum marks : 75
PART A — (5 x five = 25 marks)
ans any 5 ques..
1.Show that the set of all even permutations is a normal subgroup of .
2.If where is a prime number show that .
3.Let be an ideal of an Euclidean ring . Show that there exists such that consists exactly all where .
4.State and prove Fermat’s theorem.
5.If in a vector space has as linear span and if are linearly independent, show that we can obtain a subset of of the form consisting of linearly independent elements whose linear span is also .
6.Show that the vectors are linearly independent.
7.State and prove factor theorem.
8.If is nil potent show that is invertible if and .
PART B — (5 × 10 = 50 marks)
ans any 5 ques..
9.Let and be 2 subgroups of a group . Show that is a subgroup of if and only if .
10.Show that the number of p-Sylow subgroups in for any provided prime is of the form .
11.Show that an ideal is a maximal ideal of an Euclidean ring iff is a prime in
12.State and prove unique factorization theorem.
13.If and are vector spaces of dimensions and respectively show that is of dimension .
14.Show that is algebraic over a field if and only if is a finite extension of .
15.If is of characteristic 0 and if algebraic over , show that there exists an element such that .
16.(a) If is finite dimensional over show that is singular if and only if there exists such that .
(b) If is a characteristic root of , show that for any polynomial , is a characteristic root of .
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M.Sc. DEGREE EXAMINATION JUNE 2009.
First Year
(AY 200607 batches onwards)
Mathematics
ALGEBRA
Time : 3 hours Maximum marks : 75
PART A (5 5 = 25 marks)
Answer any FIVE questions.
1.
Show that the set of all even permutations is a normal subgroup of 
.
2.
If 
where 
is a prime number show that 
.
3.
Let 
be an ideal of an Euclidean
ring 
. Show that there exists 
such that consists exactly all 
where 
.
4. State and prove Fermats theorem.
5.
If 
in a vector space 
has 
as linear span and if 
are linearly independent,
show that we can find a subset of 
of the
form 
consisting of linearly
independent elements whose linear span is also .
6. Show that the vectors
are linearly independent.
7. State and prove factor theorem.
8.
If 
is nil potent show that 
is invertible if 
and 
.
PART B (5 10 = 50 marks)
Answer any FIVE questions.
9.
Let 
and 
be two subgroups of a group 
. Show that 
is a subgroup of if and only if 
.
10. Show
that the number of p-Sylow subgroups in for
any given prime is of the form 
.
11. Show
that an ideal 
is a maximal ideal
of an Euclidean ring iff 
is a prime in 
12. State and prove unique factorization theorem.
13. If
and are vector spaces of
dimensions 
and 
respectively show that 
is of dimension 
.
14. Show
that 
is algebraic over a field 
if and only if 
is a finite extension of .
15. If
is of characteristic 0 and if
algebraic over , show that there exists an
element 
such that 
.
16. (a) If
is finite dimensional over show that is singular if and only if
there exists 
such that 
.
(b) If 
is a characteristic
root of , show that for any polynomial
, 
is
a characteristic root of 
.
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