Exam Details
| Subject | algebra | |
| Paper | ||
| Exam / Course | m.sc. mathematics | |
| Department | ||
| Organization | acharya nagarjuna university-distance education | |
| Position | ||
| Exam Date | May, 2017 | |
| City, State | new delhi, new delhi |
Question Paper
Total No. of Questions 10] [Total No. of Pages 02
M.Sc. (Previous) DEGREE EXAMINATION, MAY 2017
First Year
MATHEMATICS
Algebra
Time 3 Hours Maximum Marks: 70
Answer any five of the following
All questions carry equal marks.
Q1) State and prove the Cayley's theorem.
If is a homomorphism of a group G into then show that
the unit element of G.
ii) −1 for all x∈G.
Q2) Suppose G is a group and that G is the internal direct product of
N1, N2, …, Nn. If T N1× N2 ×…× Nn then prove that G and T are
isomorphic.
If p is a prime number and p
then prove that G has a subgroup of
order p
.
Q3) State and prove the Cauchy's theorem for abelian groups.
Define a permutation. Resolve the permutation
1 2 3 4 5 6 7 8
1 3 2 5 6 4 8 7
into disjoint cycles. Given x
y find a permutation a such that a−1x a y.
Q4) Prove that every integral domain is a field.
Let R be a commutative ring with unit element whose only ideals are and
R itself. Then prove that R is a field.
Q5) State and prove the Fermat's theorem.
State the Eisenstein criterion, prove that the polynomial 1 x … p 1 x −
where p is prime number, is irreducible over the field of rational numbers.
Q6) If L is a finite extension of K and K is a finite extension of then prove that
L is a finite extension of F and that F].
Prove that a polynomial of degree n over a field can have atmost n roots in
any extension field.
Q7) Prove that it is impossible, by straight edge and compass alone, to trisect 60o.
Prove that K is a normal extension of a field F if and only if K is the splitting
field of some polynomial over F.
Q8) Show that a general polynomial of degree 5 is not solvable by radicals.
Prove that the fixed field of G is a sub field of K.
ii) If K is a finite extension of then is a finite group and show
that
Q9) State and prove the Schreier's theorem.
Derive the dimensionality equation d(a − d(a for
modular lattice.
Q10) Prove that every distributive lattice with more than one element can be
represented as a sub direct union of two element chains.
Define a Boolean algebra and a Boolean ring. Show that a Boolean ring can
be converted into a Boolean algebra.
M.Sc. (Previous) DEGREE EXAMINATION, MAY 2017
First Year
MATHEMATICS
Algebra
Time 3 Hours Maximum Marks: 70
Answer any five of the following
All questions carry equal marks.
Q1) State and prove the Cayley's theorem.
If is a homomorphism of a group G into then show that
the unit element of G.
ii) −1 for all x∈G.
Q2) Suppose G is a group and that G is the internal direct product of
N1, N2, …, Nn. If T N1× N2 ×…× Nn then prove that G and T are
isomorphic.
If p is a prime number and p
then prove that G has a subgroup of
order p
.
Q3) State and prove the Cauchy's theorem for abelian groups.
Define a permutation. Resolve the permutation
1 2 3 4 5 6 7 8
1 3 2 5 6 4 8 7
into disjoint cycles. Given x
y find a permutation a such that a−1x a y.
Q4) Prove that every integral domain is a field.
Let R be a commutative ring with unit element whose only ideals are and
R itself. Then prove that R is a field.
Q5) State and prove the Fermat's theorem.
State the Eisenstein criterion, prove that the polynomial 1 x … p 1 x −
where p is prime number, is irreducible over the field of rational numbers.
Q6) If L is a finite extension of K and K is a finite extension of then prove that
L is a finite extension of F and that F].
Prove that a polynomial of degree n over a field can have atmost n roots in
any extension field.
Q7) Prove that it is impossible, by straight edge and compass alone, to trisect 60o.
Prove that K is a normal extension of a field F if and only if K is the splitting
field of some polynomial over F.
Q8) Show that a general polynomial of degree 5 is not solvable by radicals.
Prove that the fixed field of G is a sub field of K.
ii) If K is a finite extension of then is a finite group and show
that
Q9) State and prove the Schreier's theorem.
Derive the dimensionality equation d(a − d(a for
modular lattice.
Q10) Prove that every distributive lattice with more than one element can be
represented as a sub direct union of two element chains.
Define a Boolean algebra and a Boolean ring. Show that a Boolean ring can
be converted into a Boolean algebra.