M- 5416 (Pages :4) Reg, No. : Name : 4 ember 202 Second Semester M.Se. Degree examination, NOV Mathematics MM 221 » ABSTRACT ALGEBRA (2020 Admission) Time s3 Ho Max. Marks : 75 urs PART -A ‘Answer any five questions from among the questions 1 to 8. Each question carries 3 marks 4. Show that 2/42 is isomorphic to 2Z,. 2, Leto bea homomorphism from a group G to a group G and let H bea subgroup of G. Then show that o(H) = {o(h) ihe H} is a subgroup ofG. 3. Determine the groups of order 99. 4. Find the splitting field of x* ~ x? -2 overQ 5, Let Fbe a field and let f(x) ¢ Fix}. Then show that an Lee sortie y two splitting fields of f(x) 6. Define the n” cyclotomic polynomial », (x) over Q. Frnd ¢,(x) 1X). For every positive integer, show that 9,(x) has integer coefficient INS. 8. Show that a factor group of a solvable group is solvable (5x 3 = 15 Marks) PT.0.PART - B tion carries nd B, Each ques! I questions from 9 to 13 by choosing A Answer all que’ iu ow that G OH iscyclicit en shi 9. (A) (a) LetGand H be finite cyclic groups. Th i . G and Hare relatively prime sg then prow te b) Let G be a group and Ha normal subgrouP of G. " ec . Gin=(aHfacG} § 4 subgroup under the operation = {aH lac 6 (aH)(bH) = abH. OR (B) (a) For any group G, show that G/z(G) is isomorphic to Inn (G). 6 (b) Let 0 be a homomorphism from a group G to a group G and let g be an element of G. Then prove the following (i) ¢ carries the identity of G to the identity of G (ii) @lg") = (o(g))" (in) If g =n, then jo(g) divides n (m) I o(g) = g' then 6'(g") = tre G1o(x) = g'} = Gkero. ‘ 10. (A) (a) Let ) G be a finite Group whose order is @ power of P. Then of show that Z(G) has more than one elem, it _ ent, § (b) Show that the ont ¥ Stoup of ord er 255is 7 OR 2(By (a) (b) 4) @) (b) (8) (a) (0) 12. (A) (a) (b) (B) (a) (b) Show that an integer of the form 2.n, where n is an odd number greater than 1 1s not the order of a simple group 6 Ifa finite non-Abelian simple group G has a subgroup of index n, then prove that Gis isomorphic to a subgroup of A 6 Let F be a field and f(x) a non coastant polynomial in F[x]. Then prove that there Is an extension E of F in which f(x) has a Zero. 6 Let f(x) be an irreducible polynomial over a field F. If F has characteristic O, then prove that f(x) has no multiple zores. 6 OR Let K be a finite extension field of the field E and let E be a finite extension field of the field F. Then. show that K is a finite extension field ofFand|k -F)=(K ENE: FI 6 If K is an algebraic extension of E and Eis an algebraic extension of F, then show that K is an algebraic extension of F. 6 For each divisor on of n, show that GF (P’") has a unique sub field of order P™. , Write £00) = 20 #00 +1 over z, as the product of linear factors. 6 OR Show that the group G of order 35 is cyclic. 6 Prove that sind is constructible iff cos @ is constructible. 6 M- 541613, (A) (a) (b) (B) (a) (b) Let F be a field of characteristic 0 and let f(x) € F[x]. Suppose that fix) splits in F(a, 8) where ate F and a’ e F(a. a), for j=23,..t. let E be the splitting field for f(x) over F in F (a,,a,.....@,). Then show that the Galois group Gal (E/F) is solvable. 6 Determine the irreducible factorization of x‘ - 1 over z, and 2, 6 OR Let n be a positive integer and let w = cos 2) + isin( 27), Then show that {cos 28) < Qo) 6 th Let @ be the primitive nm” root unity. Then show that Gal (Q(@)/Q) = Uin). (5x 12.= 60 Marks)