Abstract Algebra Test Examination-2011

Govt. M M College, Jessore
Department of Mathematics
3rd Year B. Sc. Honours Test Examination-2011
Subject – Abstract Algebra Code – 3731
Time – 4 hours Full Marks – 100

[ All questions are of equal value. Answer any Six questions ]

1.(a) Define an abelian group and order of an element of a group. Is Z, the set of all integers form a group under the operation multiplication ? Justify your answer.
(b) Let G = Q -{1} be the set of all rational numbers except 1. Then G forms a group with respect to the operation * defined by a * b = a + b –a b , .

2 (a) Define cyclic group . Prove that every sub-group of a cyclic group is cyclic.
(b) Prove that every finite group of prime order is cyclic and has no proper sub-group.

3 (a) Define sub-group of a group with enample.
(b) Let G be group and H is a non-empty finite subset of G. Then H is a sub-group of G if and only if .

4 (a) State and prove Langrange’s theorem. Is the converse of the theorem is true? Justify your answer with enample.
(b) Define normal sub-group with enample. If H and K are normal sub-group of G then prove that HK is also a normal sub-group of G.

5 (a) Define homomorphism on group. Prove that the kernel of a homomarphism is a normal sub-group .
(b) State and prove the third law on group isomorphism.

6 (a) Define ring , sub ring and integral domain. Prove that every finite integral domain is a field.
(b) Prove that Boolean ring is a commutative ring.

7 (a) Define field and ideal. Prove that a field has no proper ideal.
(b) Define prime ideal. Let s be an ideal of a commutative ring R with unity then R/S is an integral domain iff S is a prime ideal of R.

8 (a) Define principal ideal, prove that every ideal in Z is principal ideal.
(b) Let R and T be two ring and be a ring homomarphism. Prove that R/kerf Imf.

9 (a) Define a ring-polynomials. Show that it is a principal ideal ring.
(b) Prove that every finite extension of a field is an algebraic extension.

10 (a) State and prove the unique Factorization theorem.
(b) Every polynomial [x] of degree m >0 has at mast m distinct zeros in F.

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