Kuvempu University 2009 M.Sc Applied Mathematics Mathematics (PM-10.01) Algebra, / - - Question Paper

M.Sc, (Previous), Degree exam
August / September 2009 Directorate of Correspondence Course (Freshers) Mathematics Paper -
PM-10.01 :


Algebra
Time : three Hours Max. Marks : 80
Note : 1)Answer any 5 ques..
2)All ques. carry equal m. rks.

1.
a.State and prove Cayley's theorem.
b.State and prove Cauchy's theorem for abelian groups.
c.Show that no group of order 108 is simple (Marks:5+5+6)

2.
a.II H is a p - sylow sub group of a finite group G. Then for any xeG. Showthat x1 Hx is also p-sylow subgroup of G.
b.Compute two - sylow and three sylow subgroups of the symmetric group S4.
c.Define a solvable group. Show that every group of prime power order is solvable
(Marks:5+6+5)

3.
a.If is a ring R. x* = x for every x € R then show that R is commutative.
b.Prove that every integral domain can be imbedded in a field.
c.If R is a commutative ring with unit element and M is an ideal of R then show that M is maximal if and only if_5_ is a field. (Marks:5+6+5)

4.
a.Prove that the ring Gaussian integers is an Euclidean ring.
b.State Eisenstein criterion of irreducibility of polynomials. Use this criterion to
show that 1+x+x* + xfr\ p being prime over rationals is irreducible.
c.If R is UFD then so in R fx]. (Marks:5+5+6)

5.
a.Define vector spare and subspace. provide 1 example for each, Further show
that the intersection of 2 subspaces is a subspace and provide an example to show that the union of 2 subspaces need not be a subspace.
b.Stating necessary lemma that are used in the proof, prove that any 2 bases of a finite dimensional vector space have the identical number of elements.
c.Define quotient space if V is finite dimensional vector space and W a subspace then show that dim W £ dim V and dim _V = dim V-dim W. (Marks:5+5+6)

Earning:   Approval pending.