Karnataka State Open University (KSOU) 2011-1st Sem M.Sc Mathematics (Algebra) - Question Paper
Illllllllllllllllllllll Math 1.1
I Semester M.Sc. Mathematics Examination, May 20ll ALGEBRA
Time : 3 Hours Max. Marks : 80
Note x I) Answer any five questions.
2) All questions carry equal marks.
1. a) State and prove Lagrange theorem for finite groups.
b) Let f : G > G' be a group homomorphism. Then prove that ker f is a normal subgroup of G. Moreover prove that f is a one-one mapping if and only if ker f = {e}. (8+8)
2. a) Prove that every permutation aeSncan be expressed as a product of disjoint
cycles.
b) Prove the class equation of the group G. (8+8)
3. a) State and prove the first Sylow theorem.
b) Show that any group of order 52.72 is abelian. (8+8)
4. a) Show that any integral domain can be embedded in a field.
b) Let R be a commutative ring with identity. Then prove that R is a field if and only if the only ideals of R are {0} and R itself. (10+6)
5. a) State and prove the fundamental theorem of homomorphism for rings.
b) Let R be a commutative ring with identity. Prove that an ideal P in R is a
prime ideal if and only if R/P is an integral domain. (8+8)
6. a) Prove that in a Unique factorization domain, an element is a prime if and
only if it is irreducible.
b) Let F be a field and f(x) e F[x]. Then prove that aeF is a root of f(x) if and only if (x - a) divides f(x). (8+8)
7. a) Let W be a subspace of a finite-dimensional vector space V. Then prove that
W is finite - dimensional and dim W < dim V.
b) If F c: K c: L are fields, then prove that [L : F] = [L : K] [K : F]. (8+8)
8. a) Prove that any splitting field of a polynomial over F is a normal extension of F.
b) State and prove the Primitive Element Theorem. (8+8)
|
Attachment: |
| Earning: Approval pending. |
