Punjabi University 2008 M.Sc Mathematics ALGEBRA - Question Paper

M.Sc Mathematics University ques. Paper 2008

ALGEBRA

part –A(4×10=40)

(Any 4 Questions)

1.Show that a sub group N of G is normal if and only if the product of 2 right cosets of N in G is again a right coset of N in G.

2.State and Prove cayley’s theorem.

3.Derive class formula for finite groups.

4.Indentify and establish the prime ideals in the ring of integers.

5.Prove that the ring of Gaussiom integers is an Euclidean ring.

6.State and prove Eisenstein Criterion.

7.Show that an element a in a field K is alegebraic over F if and only if F(a) is a finite extension of F.

8.For every prime number p, and every positive integer m, prove that there exists a unique field having Pm elements.


part –B (3×20=60)

(Any 3 Questions)

9.State and prove 1st and 2nd Sylow theorems.

10.(a) describe solvable group.Give an example.

(b) If G is a group and N is a normal subgroup of G such that both N and G are solvable,prove that G is solvable.

11.(a) Prove that every Euclidean ring contains a unit element.

(b) State and prove Unique factorization theorem.

12.If R is a unique factorization domain, Prove that R(x) is a unique factorization domain.

13.(a) Prove that is K a normal extension of F if and only if K is the splitting field of a few polynomial over F.

(b) If K is a finite extension of F, prove that G(K,F) is a finite group and aG (K,F) = [K:F].

14.(a) If P(x) EF(x) is solvable by radicals over F, Prove that the Galois group over F of P(x) is solvable group.

(b)Show that ,If F is a finite field and a ? 0, ß ? 0 are 2 elements of F, them we can obtain elements a and ß is F such that 1+ aa²+ ßb²=0.

Earning:   Approval pending.