BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination December, 2014 ELECTIVE COURSE : MATHEMATICS MTE-06 : ABSTRACT ALGEBRA

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1 072 A0.2 No. of Printed Pages : 8 BACHELOR'S DEGREE PROGRAMME (BDP) Term-End Examination December, 2014 ELECTIVE COURSE : MATHEMATICS MTE-06 : ABSTRACT ALGEBRA MTE-06 Time : 2 hours Maximum Marks : 50 (Weightage : 70%) Note : Attempt five questions in all. Question no. 7 is compulsory. Answer any four questions from Q. No. 1 to 6. Use of calculators is not allowed. 1. (a) Express the permutation (b) (c) CT = ' , as a product of disjoint cycles and as a product of transpositions. Find a permutation i in S9 such that TOO' = (TOT = e, where e is the identity permutation in S9. 4 Give examples each of a finite and an infinite integral domain. 2 Defining addition and multiplication in R2 component-wise, i.e. (x1, y 1 ) + (x2, y 2 ) = (x 1 + x 2' y 1 + y 2 ) (x1, y 1 ). (x 2' y 2 ) = (x1x2, y 1 y 2 ) prove that R2 is a ring. Is it an integral domain? Justify your answer. 4 MTE-06 1 P.T.O.

2 2. (a) If R is a ring such that x2 = x, for every x E R, show that R is a commutative ring. Give an example of such a ring. 4 (b) Find the nil radical of an integral domain. 2 (c) If F is a field with 49 elements, prove that x 49 = x, V x E F. Also, find the characteristic of F (a) Show that Q + F5- Q is a subfield of C. Also, check that it is the quotient field of Z + V16 Z. 4 (b) Classify all the groups of order less than or equal to 6 upto isomorphism. 4 (c) Prove that the polynomial 6x5+ 30x 4-40x x + 40 is irreducible in Q [x]. Is it irreducible in Z [x]? 2 4. (a) Let b 0 d a, b, d E R, ad * 0 and H. x _ 0 1 x E R Show that H is a subgroup of G. Further, show that H is a normal subgroup of G. Is G/H abelian? Justify your answer. 5 MTE-06 2

3 (b) Show that f : Z + iz -p Z2 defined by + ib) = (a b) (mod 2) is an onto ring homomorphism. Determine ker f. Is it a maximal ideal? Justify your answer (a) Let S = {(x, y) I x, y E R) be the plane in the rectangular co-ordinate system. Define a relation by (x1, y1)n (x2, y2) iff x1 x2 is an integer. (i) Show that N is an equivalence relation. (ii) Give a geometric description of the equivalence class to which (0, 0) belongs. 4 (b) If G is a group of even order, prove that it has an element a e satisfying a2 = e, where e is the identity element of G. 2 (c) Show that the set G of matrices of the form cos a sin a sin a cos a where a is a real number, forms a group under matrix multiplication. 4 MTE-06 3 P.T.O.

4 6. (a) Does the ring Z7[x] Kx2 + 8) have nilpotent elements? Justify your answer. 3 (b) Find all the maximal ideals of the ring Z36. 4 (c) Let M2(Z) be the ring of all 2 x 2 matrices over the integers and let - aa b R= a, b E Z a b a Is R a subring of M2(Z)? Justify your answer Which of the following statements are true and which are false? Give reasons for your answer. 10 (i) If G is a group of order n and if d is a divisor of n, then there exists a subgroup of G of order d. (ii) The ring Q of all rational numbers has proper non-trivial subrings, but has no proper non-trivial ideals. (iii) If every subgroup of a group G is normal, then G is abelian. (iv) There exists a field with 100 elements. (v) Any polynomial of degree n over a ring R can have at most n roots in the ring R. MTE-06 4

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