CSIR Algebra Problems PDF

CSIR - Algebra Problems with solutions N. Annamalai DST - INSPIRE Fellow (SRF) Department of Mathematics Bharathidasan University Tiruchirappalli -620024 E-mail: [email protected]...

CSIR - Algebra Problems with solutions N. Annamalai DST - INSPIRE Fellow (SRF) Department of Mathematics Bharathidasan University Tiruchirappalli -620024 E-mail: [email protected] Website: https://annamalaimaths.wordpress.com ————————————————————————————————— 1. [June 2009 - B] Let α = (1, 3, 5, 7, 9, 11) and β = (2, 4, 6, 8) be two permu- tations in S100. Then the order of αβ is (i) 4 (ii) 6 (iii) 12 (iv) 100. 2. [June 2009 - B] The number of groups of order 121, up to isomorphism, is (i) 1 (ii) 2 (iii) 11 (iv) 10. 3. [June 2009 - B] Let G a non-abelian group of order 21. Let H be a Sylow 3-subgroup and K be a Sylow 7-subgroup of G. Then (i) H and K are both normal in G (ii) H is normal but K is NOT normal in G (iii) K is normal but H is NOT normal in G (iv) Neither H nor K is normal in G. Z5 [ x ] 4. [June 2009 - B] Let f ( x ) ∈ Z5 [ x ] be a polynomial such that < f ( x )> is a field. Then one of the choices for f ( x ) is (i) x2 + 1 (ii) x2 + 3 (iii) x3 + 1 (iv) x3 + 3. 5. [Dec 2009 - B] Let R be a commutative ring. Let I and J be ideals of R. Let I − J = { x − y | x ∈ I, y ∈ J } and I J = { xy | x ∈ I, y ∈ J }. Then (i) I − J is an ideal and I J is an ideal in R (ii) I − J is an ideal and I J need not be an ideal in R (iii) I − J need not be an ideal but I J is an ideal in R (iv) Neither I − J nor I J need to be an ideal in R. 6. [Dec 2009 - B] The polynomial x3 + 5x2 + 5 is (i) irreducible over Z but reducible over Z5 (ii) irreducible over both Z and Z5 (iii) reducible over Z but irreducible over Z5 (iv) reducible over both Z and Z5. 7. [Dec 2009 - B] In Z[i ] (i) 5 and 6 are irreducible 1  (ii) 5 is irreducible but 6 is reducible (iii) 5 is reducible but 6 is irreducible (iv) neither 5 nor 6 is irreducible. 8. [Dec 2009 - B] The number of distinct homomorphisms from Z12 to Z25 is (i) 1 (ii) 2 (iii) 3 (iv) 4. 9. [Dec 2009 - B] The number of 5-Sylow subgroups of S6 is (i) 16 (ii) 6 (iii) 36 (iv) 1. 10. [Dec 2009 - B] Let G be a group of order 14 such that G is not abelian. Then the number of elements of order 2 in G is equal to (i) 7 (ii) 6 (iii) 13 (iv) 2. 11. [June 2010 - B] The number of generators of a cyclic group of order 12 is (i) 1 (ii) 2 (iii) 3 (iv) 4. 12. [June 2010 - B] Up to an isomorphism, the number of groups of order 33 is (i) 3 (ii) 11 (iii) 1 (iv) Infinitely many. 13. [June 2010 - B] Let R = Q[ x ]. Let I be the principal ideal < x2 + 1 > and J be the principal ideal < x2 >. Then (i) R/I is a field and R/J is a field (ii) R/I is an integral domain and R/J is a field (iii) R/I is a field and R/J is a PID (iv) R/I is a field and R/J is not an integral domain. 14. [June 2010 - B] The polynomial ring Z[ x ] is (i) a Euclidean domain but not a PID (ii) a PID but not Euclidean (iii) Neither PID nor Euclidean (iv) both PID and Euclidean. 15. [June 2010 - B] The polynomial x3 − 7x2 + 15x − 9 is (i) irreducible over both Z and Z3 (ii) irreducible over Z but reducible over Z3 (iii) reducible over Z but irreducible over Z3 (iv) reducible over both Z and Z3. 16. [June 2010 - B] A permutation a of {1, 2, · · · , n} is called a derangement if α(i ) 6= i for every i. Let dn denote the number of derangements of {1, 2, · · · , n}. Then d4 is equal to (i) 3 (ii) 9 (iii) 12 (iv) 24. 17. [June 2010 - B] The number of sub fields of a finite field of order 310 is equal to (i) 4 (ii) 5 (iii) 3 (iv) 10. 18. [June 2011 - B] The unit digit of 2100 is (i) 2 (ii) 4 (iii) 6 (iv) 8. 2  √ √ 19. [June √ 2011 - B] The degree of the extension Q ( 2 + 3 2) over the field Q( 2) is (i) 1 (ii) 2 (iii) 3 (iv) 6. 20. [June 2011 - B] Consider a group G. Let Z ( G ) be its centre. For n ∈ N, define Jn = {( g1 , g2 , · · · , gn ) ∈ Z ( G ) × Z ( G ) × · · · × Z ( G ) : g1 g2 · · · gn = e}. As a subset of a direct product groups G × · · · × G (n times direct product of the group G), Jn is (i) not necessarily subgroup (ii) a subgroup but not necessarily a normal subgroup (iii) normal subgroup (iv) isomorphic to the direct product Z ( G ) × · · · × Z ( G ) (n − 1 times). 21. [June 2011 - B] Let I1 be the ideal generated by x4 + 3x2 + 2 and I2 be the ideal generated by x3 + 1 in Q[ x ]. If F1 = Q[ x ]/I1 and F2 = Q[ x ]/I2 , then (i) F1 and F2 are fields (ii) F1 is a field but F2 is not a field (iii) F1 is not a field while F2 is a field (iv) neither F1 nor F2 is a field. 22. [June 2011 - B] Let G be group of order 77. Then the centre of G is isomorphic to (i) Z1 (ii) Z7 (iii) Z11 (iv) Z77 23. [June 2011 - C] Let G = Z10 × Z15. Then (i) G contains exactly one element of order 2 (ii) G contains exactly 5 element of order 3 (iii) G contains exactly 24 element of order 5 (iv) G contains exactly 24 element of order 10. 24. [June 2011 - C] Let H = {e, (1, 2), (3, 4)} and K = {e, (1, 2), (3, 4), (1, 3), (2, 4), (1, 4), (2, 3)} be subgroups of S4. Then (i) H and K are normal subgroups of S4 (ii) H is normal in K and K is normal in A4 (iii) H is normal in A4 but not normal in S4 (iv) K is normal in S4 but H is not. 25. [June 2011 - C] Let < p( x ) > denote the ideal generated by the polyno- mial p( x ) in Q[ x ]. If f ( x ) = x3 + x2 + x + 1 and g( x ) = x3 − x2 + x − 1, then (i) < f ( x ) > + < g( x ) >=< x3 + x > (ii) < f ( x ) > + < g( x ) >=< f ( x ) g( x ) > (iii) < f ( x ) > + < g( x ) >=< x2 + 1 > (iv) < f ( x ) > + < g( x ) >=< x4 − 1 >. 26. [June 2011 - C] Let I1 be the ideal generated by x2 + 1 and I2 be the ideal generated by x3 − x2 + x − 1 in Q[ x ]. If R1 = Q[ x ]/I1 and R2 = Q[ x ]/I2 , then (i) R1 and R2 are fields (ii) R1 is a field but R2 is not a field (iii) R1 is an integral domain but R2 is not an integral domain (iv) R1 and R2 are not integral domains. 3 27. [Dec 2011 - B] Let p be a prime number. The order of a p-sylow subgroup of the group GL50 (F p ) of invertible 50 × 50 matrices with entries from the finite fieldF p , equals: (i) p50 (ii) p125 (iii) p1250 (iv) p1225 28. [Dec 2011 - B] The number of multiples of 1044 that divide 1055 is (i) 11 (ii) 12 (iii) 121 (iv) 144. 29. [Dec 2011 - B] The number of group homomorphisms from the symmet- ric group S6 to Z6 is (i) 1 (ii)2 (iii)3 (iv) 6. 30. [Dec 2011 - C] Let Z[i ] denote the ring of Gaussian integers. For which of the following values of n is the quotient ring Z[i ]/nZ[i ] an integral domain? (i) 2 (ii) 13 (iii) 19 (iv) 7. 31. [Dec 2011 - C] Which of the following integral domains are Euclidean domains? √ Z[ x ]   (i) Z[ −3] (ii) Z[ x ] (iii) R[ x2 , x3 ] (iv) (2,x) [y]. 32. [Dec 2011- C] Let G be the Galois group of the splitting field of x5 − 2 over Q. Then, which of the following statements are true? (i) G is cyclic (ii) G is non-abelian (iii) the order of G is 20 (iv) G has an element of order 4. 33. [June 2012 - B] The number of positive divisors of 50,000 is (i) 20 (ii) 30 (iii) 40 (iv) 50. √ 34. [June 2012 - B] The number 2eix is (i) a rational number (ii) a transcendental number (iii) an irrational number (iv) an imaginary number. 35. [June 2012 - B] Let F be a field of 8 elements and A = { x ∈ F : x7 = 1 and x k 6= 1 for all natural number k < 7}. Then the number of ele- ments of A is (i) 1 (ii) 2 (iii) 3 (iv) 6. 36. [June 2012 - B] Consider the group G = Q/Z. Let n be a positive integer. Then is there a cyclic subgroup of order n? (i) not necessarily (ii) yes, a unique one (iii) yes, but not necessarily unique one (iv) never. 37. [June 2012 - B] Let f ( x ) = x3 + 2x2 + 1 and g( x ) = 2x2 + x + 2. Then over Z3 (i) f ( x ) and g( x ) are irreducible (ii) f ( x ) is irreducible but g( x ) is not (iii) g( x ) is irreducible but f ( x ) is not (iv) neither f ( x ) not g( x ) is irreducible. 38. [June 2012 - B] The number of non-trivial ring homomorphisms from Z12 to Z28 is (i) 1 (ii) 3 (iii) 4 (iv) 7. 4 39. [June 2012 - C] Let R = Q/I where I is the ideal generated by 1 + x2. Let y be a coset of x in R. Then (i) 1 + y2 is irreducible over R (ii) 1 + y + y2 is irreducible over R (iii) y2 − y + 1 is irreducible over R (iv) 1 + y + y2 + y3 is irreducible over R. 40. [June 2012 - C] which of the following is true? (i) sin 7o is algebraic over Q (ii) cos √ π/17 is algebraic over Q − √ (iii) sin 1 is algebraic over Q (iv) 2 + π is algebraic over Q()π. 1 41. [June 2012 - C] Let f ( x ) = x3 + x2 + x + 1 and g( x ) = x3 + 1. Then in Q[ x ], (i) gcd( f ( x ), g( x )) = x + 1 (ii) gcd( f ( x ), g( x )) = x2 − 1 (iii) lcm( f ( x ), g( x )) = x5 + x3 + x2 + 1 (iv) lcm( f ( x ), g( x )) = x5 + x4 + x3 + x2 + 1. 42. [June 2012 - C] For any group G of order 36 and any subgroup H of G order 4, (i) H ⊂ Z ( G ) (ii) H = Z ( G ) (iii) H is normal in G (iv) H is an abelian group. 43. [June 2012 - C] Let G denote the group S4 × S3. Then (i) a 2-sylow subgroup of G is normal (ii) a 3-sylow subgroup of G is normal (iii) G has a nontrivial normal subgroup (iv) G has a normal subgroup of order 72. 44. [Dec 2012 - B] The last two digits of 781 are (i) 07 (ii) 17 (iii) 37 (iv) 47. 45. [Dec 2012 - B] In which of the following fields, the polynomial x3 − 312312x + 123123 is irreducible in F[ x ]? (i) the field F3 with 3 elements (ii) the field F7 with 7 elements (iii) the field F13 with 13 elements (iv) the field Q of rational numbers. 46. [Dec 2012 - B] let ω be a complex √ number such√that ω 3 = 1 and ω 6= 1. Suppose L is the field Q( 2, ω ) generated by 2 and ω over the field 3 3 Q of rational numbers. Then the number of subfields K of L such that Q ⊆ K ⊆ L is (i) 2 (ii) 3 (iii) 4 (iv) 5. 47. [Dec 2012 - C] For any positive integer m, let φ(m) denote the number of integers k such that ≤ k ≤ m and GCD (k, m) = 1. Then which of the following statements are necessarily true? (i) φ(n) divides n for every positive integers n (ii) n divides φ( an − 1) for every positive integers a and n (iii)n divides φ( an − 1) for every positive integers a and n such that GCD ( a, n) = 1 5  (iv) a divides φ( an − 1) for every positive integers a and n such that GCD ( a, n) = 1. 48. [Dec 2012 - C] For a positive integer n ≥ 4 and a prime number p ≤ n, U p,n denote the union of all p-sylow subgroups of a alternating group An on n letters. Also let K p,n denote a subgroup of An generated by U p,n , and let |K p,n | denote the order of K p,n. Then (i) |K2,4 | = 12 (ii) |K2,4 | = 4 (iii) |K2,5 | = 60 (iv) |K3,5 | = 30. 49. [Dec 2012 - C] For a positive integer n, let f n ( x ) = x n−1 + x n−2 + · · · + x + 1. Then (i) f n ( x ) is an irreducible polynomial in Q[ x ] for every positive integer n. (ii) f p ( x ) is an irreducible polynomial in Q[ x ] for every prime integer p. (iii) f pe ( x ) is an irreducible polynomial in Q[ x ] for every prime number p and for every positive integer e. e −1 (iv) f p ( x p ) is an irreducible polynomial in Q[ x ] for every prime num- ber p and for every positive integer e. √ √ 50. [Dec 2012 - C] Consider the √ ring R = Z[ −5] = { a + b −5 : a, b ∈ Z} and the element α = 3 + −5 of R. Then (i) α is prime (ii) α is irreducible (iii) R is not a unique factorization domain (iv) R is not an integral domain. 51. [Dec 2102 - C] Consider the polynomial f ( x ) = x4 − x3 + 14x2 + 5x + 16. Also for a prime number p, let F p denote the field with p elements. Which of the following are always true? (i) Considering f as a polynomial with coefficients in F3 , it has no roots in F3. (ii) Considering f as a polynomial with coefficients in F3 , it is a product of two irreducible factors of degree 2 over F3. (iii) Considering f as a polynomial with coefficients in F7 , it has an irreducible factor of degree 3 over F7. (iv) f is a product of two polynomials of degree 2 over Z. 52. [Dec 2012 - C] For a positive integer m, let am denote the number of Q[ x ] distinct prime ideals of the ring < xm −1>. Then (i) a4 = 2 (ii) a4 = 3 (iii) a5 = 2 (iv) a5 = 3. 53. [June 2013 - B] Let G be a simple group of order 168. What is the number of subgroups of G of order 7? (i) 1 (ii) 7 (iii) 8 (iv) 28. 54. [June 2013 - B] What is the smallest positive integer in this set {24x + 60y + 2000z | x, y, z ∈ Z}? (i) 2 (ii) 4 (iii) 6 (iv) 24. 55. [June 2013 - B] Which of the following ring is PID? (i) Q[ x, y]/ < x > (ii) Z ⊕ Z (iii) Z[ x ] (iv) M2 (Z) the ring of all 2 × 2 matrices with entries in Z. 6 56. [June 2013 - B] Let F ⊆ C be the splitting field of x7 − 2 over Q, and z = e2πi/7 √ , a primitive seventh root of unity. Let [ F : Q(z)] = a and [ F : Q( 2)] = b. Then 7 (i) a = b = 7 (ii) a = b = 6 (iii) a > b (iv) a < b. 57. [June 2013 - C] Let σ = (12)(345) and τ = (123456) be permutations in S6. Which of the following statements are true? (i) The subgroups < σ > and < τ > are isomorphic to each other (ii) σ and τ are conjugate in S6 (iii) < σ > ∩ < τ > is a trivial group (iv) σ and τ commute. 58. [June 2013 - C] Let Sn denote the symmetric group on n symbols. The group S3 ⊕ (Z/2Z) is isomorphic to which of the following groups? (i) Z/12Z (ii) Z/6Z ⊕ Z/2Z (iii) A4 , the alternating group of order 12 (iv) D6 , the dihedral group of order 12. 59. [June 2013 - C] Let F = F3 [ x ]/( x3 + 2x − 1), where F3 is the field with 3 elements. Which of the following statements are true? (i) F is a field with 27 elements (ii) F is separable but not a normal extension of F3 (iii) The automorphism group of F is cyclic (iv) The automorphism group of F is abelian but not cyclic. 60. [June 2013 - C] Which of the following polynomials are irreducible over the given rings? (i) x5 + 3x4 + 9x + 15 over Q (ii) x3 + 2x2 + x + 1 over Z7 (iii) x3 + x2 + x + 1 over Z (iv) x4 + x3 + x2 + x + 1 over Z. 61. [Dec 2013 - B] How many normal subgroups does a non-abelian group G of order 21 have other than {e} and G? (i) 0 (ii) 1 (iii) 3 (iv) 7. 62. [Dec 2013 - B] The number of group homomorphisms from the symmet- ric group S3 to the additive group Z6 is (i) 1 (ii) 2 (iii) 3 (iv) 0. 63. [Dec 2013 - C] Determine which of the following cannot be the class equation of a group (i) 10 = 1 + 1 + 1 + 2 + 5 (ii) 4 = 1 + 1 + 2 (iii) 8 = 1 + 1 + 3 + 3 (iv) 6 = 1 + 2 + 3. 64. [Dec 2013 - C] Let F and F 0 be two finite fields order q and q0 respectively. Then (i) F 0 contains a subfield isomorphic to F if and only if q ≤ q0 (ii) F 0 contains a subfield isomorphic to F if and only if q divides q0 7  (iii) If the gcd of q and q0 is not 1, then both are isomorphic to subfields of some finite field L (iv) Both F and F 0 are quotient rings of the ring Z[ x ]. 65. [Dec 2013 - C] Let R be a non-zero commutative ring with unity 1R. Define the characteristic of R to be the order 1R in ( R, +) if it is finite and to be zero if the order of 1R in ( R, +) is infinite. We denote the characteristic of R by char ( R). In the following, let R and S be non-zero commutative ring with unity. Then (i) char ( R) is always a prime number. (ii) if S is a quotient ring of R, then either char (S) divides char ( R), or char (S) = 0 (iii) if S is a subring of R containing 1R then char (S) = char ( R) (iv) if char ( R) is a prime number, then R is a field. 66. [Dec 2013 - C] Let R be the ring obtained by taking quotient of (Z/6Z)[ x ] by the principal ideal (2x + 4). Then (i) R has infinitely many elements (ii) R is field (iii) 5 is a unit R (iv) 4 is a unit in R. 67. [Dec 2013 - C] Let f ( x ) = x3 + 2x2 + x − 1. Determine in which of the case f is irreducible over the field K. (i) K = Q, the field of rational numbers (ii) K = R, the field of real numbers (iii) K = F2 , the finite field of 2 elements (iv) K = F3 , the finite field of 3 elements. 68. [June 2014 - B] The total number of non-isomorphic groups of order 121 is (i) 2 (ii) 1 (iii) 61 (iv) 4. 69. [June 2014 - B] Let G denote the group of all automorphisms of the field F3100 that consists of 3100 elements. Then the number of distinct subgroups of G is equal to (i) 4 (ii) 3 (iii) 100 (iv) 9. 70. [June 2014 - B] Let p, q be distinct primes. Then (i) Z/p2 qZ has exactly 3 distinct ideals (ii) Z/p2 qZ has exactly 3 distinct prime ideals (iii) Z/p2 qZ has exactly 2 distinct prime ideals (iv) Z/p2 qZ has a unique maximal ideal. 71. [June 2014 - C] Let f ( x ) = x4 + 3x3 − 9x2 + 7x + 27 and let p be prime. Let f p ( x ) denote the corresponding polynomial with coefficients in Z p. Then (i) f 2 ( x ) is irreducible over Z2 (ii) f ( x ) is irreducible over Q (iii) f 3 ( x ) is irreducible over Z3 (iv) f ( x ) is irreducible over Z. 8 72. [June 2014 - C] Suppose ( F, +, ·) is the finite field with 9 elements. Let G = ( F, +) and H = ( F {0}, ·) denote the underlying additive and multiplicative groups respectively. Then (i) G ∼= Z/3Z × Z/3Z (ii) G ∼ = Z/9Z (iii) H ∼ = Z/2Z × Z/2Z × Z/2Z (iv) G ∼= Z/3Z × Z/3Z and H ∼ = Z/8Z. 73. [June 2014 - C] Consider the multiplicative group G of all the (complex) 2n -th roots of unity where n = 0, 1, 2, · · ·. Then (i) Every proper subgroup of G is finite (ii) G has a finite set of generators (iii) G is cyclic (iv) Every finite subgroup of G is cyclic. 74. [June 2014 - C] Let R be the ring of all entire functions. Then (i) The units in R are precisely the nowhere vanishing entire functions. (ii) The irreducible elements of R are, up to multiplication by a unit, a linear polynomial of the form z − α, where α ∈ C (iii) R is an integral domain (iv) R is a unique factorization domain. 75. [June √2014 - C] Pick the correct statements. (i) Q( √2) and Q(i ) are isomorphic as Q-vector spaces (ii) Q( 2) and√ Q(i ) are isomorphic as fields (iii) GalQ (Q( 2)/Q) ∼ = GalQ (Q(i )/Q) √ (iv) Q( 2) and Q(i ) are both Galois extensions of Q. √ √ √ 76. [Dec 2014 - B] Find the degree of the field extension Q( 2, 4 2, 8 2) over Q. (i) 4 (ii) 8 (iii) 14 (iv) 32. 77. [Dec 2014 - B] Let G be a Galois group of a field with 9 elements over its subfield with three elements. Then the number of orbits for the action of G on the field with 9 elements is (i) 3 (ii) 5 (iii) 6 (iv) 9. 78. [Dec 2014 - B] The number of conjugacy classes in the permutation group S6 is (i) 12 (ii) 11 (iii) 10 (iv) 6. 79. [Dec 2014 - B] In the group of all invertible 4 × 4 matrices with entries in the field of 3 elements, any 3-sylow subgroup has cardinality (i) 3 (ii) 81 (iii) 243 (iv) 729. 80. [Dec 2014 - C] Let G be a non-abelian group. Then, its order can be: (i) 25 (ii) 55 (iii) 125 (iv) 35. 81. [Dec 2014 - C] Let R[ x ] be the polynomial ring over R in one variable. Let I ⊆ R[ x ] be an ideal. Then (i) I is a maximal ideal iff I is anon-zero prime ideal (ii) I is a maximal ideal iff the quotient ring R[ x ]/I is isomorphic to R 9  (iii) I is a maximal ideal iff I = ( f ( x )) where f ( x ) is a non-constant irreducible polynomial over R (iv) I is a maximal ideal iff there exist a non-constant polynomial f ( x ) ∈ I of degree ≤ 2. 82. [Dec 2014 - C] Let G be a group of order 45. Then (i) G has an element of order 9 (ii) G has a subgroup of order 9 (iii) G has a normal subgroup of order 9 (iv) G has a normal subgroup of order 5. 83. [Dec 2014 - C] Which of the following is/are true? (i) Given any positive integer n, there exists a field extension of Q of degree n (ii) Given any positive integer n, there exist fields F and K such that F ⊆ K and K is Galois over F with [K : F ] = n (iii) Let K be Galois extensions of Q with [K : Q] = 4. Then there is a field L such that K ⊇ L ⊇ Q, [ L : Q] = 2 and L is a Galois extension of Q (iv) There is an algebraic extension K of Q such that [K : Q] is not finite. 84. [June 2015 - B] The number of subfields of a field of cardinality 2100 is (i)2 (ii) 4 (iii) 9 (iv) 100. 85. [June 2015 - B] Up to isomorphism, the number of abelian groups of order 108 is (i) 12 (ii) 9 (iii) 6 (iv) 5. 86. [June 2015 - B] Let R be the ring Z[ x ]/(( x2 + x + 1)( x3 + x + 1)) and I be the ideal generated by 2 in R. What is the cardinality of the ring R? (i) 27 (ii) 32 (iii) 64 (iv) Infinite. 87. [June 2015 - C] Let σ : {1, 2, 3, 4, 5} → {1, 2, 3, 4, 5} be a permutation such that σ−1 ( j) ≤ σ ( j) ∀ j, 1 ≤ j ≤ 5. Then which of the following are true? (i) σ ◦ σ ( j) = j for all j, 1 ≤ j ≤ 5 (ii) σ−1 ( j) = σ ( j) for all j, 1 ≤ j ≤ 5 (iii) The set {k | σ (k) 6= k} has an even number of elements (iv) The set {k | σ (k) = k } has an odd number of elements. 88. [June 2015 - C] If x, y and z are elements of a group such that xyz = 1, then (i) yzx = 1 (ii) yxz = 1 (iii) zxy = 1 (iv) zyx = 1. 89. [June 2015 - C] Which of the following cannot be the class equation of a group of orde 10? (i) 1 + 1 + 1 + 2 + 5 = 10 (ii) 1 + 2 + 3 + 4 = 10 (iii) 1 + 2 + 2 + 5 = 10 (iv) 1 + 1 + 2 + 2 + 2 + 2 = 10. 90. [June 2015 - C] Let C [0, 1] be the ring of all continuous functions on [0, 1]. Which of the following statements are true? (i) C [0, 1] is an integral domain 10  (ii) The set of all functions vanishing at 0 is maximal ideal. (iii) The set of all functions vanishing at both 0 and 1 is a prime ideal. (iv) If f ∈ C [0, 1] is such that ( f ( x ))n = 0 for all x ∈ [0, 1] for some n > 1, then f ( x ) = 0 for all x ∈ [0, 1]. 91. [June 2015 - C] Determine Which of the following polynomials are irre- ducible over the indicated rings. (i) x5 − 3x4 + 2x3 − 5x + 8 over R (ii) x3 + 2x2 + x + 1 over Q (iii) x3 + 3x2 − 6x + 3 over Z (iv) x4 + x2 + 1 over Z/2Z. 92. [Dec 2015 - B] A group G is generated by the elements x, y with the relations x3 = y2 = ( xy)2 = 1. The order of G is (i)4 (ii) 6 (iii) 8 (iv) 12. 93. [Dec 2015 - B] Let R be a Euclidean domain such that R is not a field. Then the polynomial ring R[ x ] is always (i) a Euclidean domain. (ii) a principal ideal domain, but not a Euclidean domain. (iii) a unique factorization domain, but not a principal ideal domain. (iv) not a unique factorization domain. 94. [Dec 2015 - B] Which of the following is an irreducible factor of x12 − 1 over Q? (i) x8 + x4 + 1 (ii) x4 + 1 (iii) x4 − x2 + 1 (iv) x5 − x4 + x3 − x2 + x − 1. 95. [Dec 2015 - C] Let an denote the number of those permutations σ on {1, 2, · · · , n} such that σ is a product of exactly two disjoint cycles. Then: (i) a5 = 50 (ii) a4 = 14 (iii) a5 = 40 (iv) a4 = 11. 96. [Dec 2015 - C] Let G be a simple group of order 60. Then (i) G has six Sylow-5 subgroups (ii) G has four Sylow-3 subgroups. (iii) G has a cyclic subgroup of order 6. (iv) G has a unique element of order 2. 97. [Dec 2015 - C] Let A denote the quotient ring Q[ x ]/( x3 ). Then (i) There are exactly three distinct proper ideals in A (ii) There is only one prime ideal in A (iii) A is an integral domain (iv) Let f , g be in Q[ x ] such that f¯ · ḡ = 0 in A. Here f¯ and ḡ denote the image of f and g respectively in A. Then f (0) · g(0) = 0. 98. [Dec 2015 - C] Which of the following quotient rings are fields? (i) F3 [ x ]/( x2 + x + 1), where F3 is the finite field with 3 elements. (ii) Z[ x ]/( x − 3) (iii) Q[ x ]/( x2 + x + 1) (iv) F2 [ x ]/( x2 + x + 1), where F2 is the finite field with 2 elements. 99. [Dec 2015 - C] Let ω = cos 2π 10 + i sin 10. Let K = Q( ω ) and let L = 2π 2 Q(ω ). Then (i) [ L : Q] = 10 (ii) [ L : K ] = 2 (iii) [K : Q] = 4 (iv) L = K. 11 100. [Dec 2015 - C] For n ≥ 1, let (Z/nZ)∗ be the group of units of Z/nZ. Which of the following groups are cyclic? (i) (Z/10Z)∗ (ii) (Z/23 Z)∗ (iii) (Z/100Z)∗ (iv) (Z/163Z)∗. 101. [June 2016 - B] Let p be a prime number. How many distinct sub-rings (with unity) of cardinality p does the field F p2 have? (i) 0 (ii) 1 (iii) p (iv) p2. 102. [June 2016 - B] Let G = (Z/25Z)∗ be the group of units (i.e. the elements that have a multiplicative inverse) in the ring (Z/25Z). Which of the following is a generator of G? (i) 3 (ii) 4 (iii) 5 (iv) 6. 103. [June 2016 - B] Let p ≥ 5 be a prime. Then (i) F p × F p has at least five subgroups of order p (ii) Every subgroup of F p × F p is of the form H1 × H2 where H1 , H2 are subgroups F p (iii) Every subgroup of F p × F p is an ideal of the ring F p × F p (iv) The ring F p × F p is a field. 104. [June 2016 - C] Let G be a finite abelian group of order n. Pick each correct statement from below. (i) If d divides n, there exists a subgroup of G of order d (ii) If d divides n, there exists an element of order d in G (iii) If every proper subgroup of G is cyclic, then G is cyclic (iv) If H is a subgroup of G, there exists a subgroup N of G such that G/N ∼ = H. 105. [June 2016 - C] Consider the symmetric group S20 and its subgroup A20 consisting of all even permutations. Let H be a 7-Sylow subgroup of A20. Pick each correct statement from below: (i) | H | = 49 (ii) H must be cyclic (iii) H is a normal subgroup of A20 (iv) Any 7-Sylow subgroup of S20 is a subset of A20. 106. [June 2016 - C] Let pbe a prime. Pick each correct statement from below. Up to isomorphism, (i) there are exactly two abelian groups of order p2 (ii) there are exactly two groups of order p2 (iii) there are exactly two commutative rings of order p2 (iv) there is exactly one integral domain of order p2. 107. [June 2016 - C] Let R be a commutative ring with unity, such that R[ x ] is a UFD. Denote the ideal ( x ) of R[ x ] by I. Pick each correct statement from below: (i) I is prime ideal (ii) If I is maximal, then R[ x ] is a PID (iii) If R[ x ] is a Euclidean domain, then I is maximal (iv) If R[ x ] is a PID, then it is a Euclidean domain. 12 108. [June 2016 - C] Let f ( x ) ∈ Z[ x ] be a polynomial of degree ≤ 2. Pick each correct statement from below (i) If f ( x ) is irreducible in Z[ x ], then it is irreducible in Q[ x ] (ii) If f ( x ) is irreducible in Q[ x ], then it is irreducible in Z[ x ] (iii) If f ( x ) is irreducible in Z[ x ], then for all primes p the reduction f (¯x ) of f ( x ) modulo p is irreducible in F p [ x ] (iv) If f ( x ) is irreducible in Z[ x ], then it is irreducible in R[ x ]. 109. [Dec 2016 - B] Let Sn denote the permutation group on n symbols and An be the subgroup of even permutations. Which of the following is true? (i) There exists a finite group which is not a subgroup of Sn for any n ≥ 1. (ii) Every finite group is a subgroup of An for some n ≥ 1. (iii) Every finite group is a quotient of An for some n ≥ 1. (iv) No finite abelian group is a quotient of Sn for n > 3. 110. [Dec 2016 - B] What is the number of non-singular 3 × 3 matrices over F2 , the finite field with two elements? (i) 168 (ii) 384 (iii) 23 (iv) 32. 111. [Dec 2016 - C] Consider the following subsets of the group of 2 × 2 non- singular  over R :  matrices     a b 1 b G= | a, b, d ∈ R, ad = 1 , H = |b∈R. 0 d 0 1 Which of the following statements are correct? (i) G forms a group under matrix multiplication. (ii) H is a normal subgroup of G. (iii) The quotient group G/H is well-defined and is Abelian. (iv) The quotient group G/H is well defined and is isomorphic to the group of 2 × 2 diagonal matrices (over R) with determinant 1. 112. [Dec 2016 - C] Let C be the field of complex numbers and C∗ be the group of non zero complex numbers under multiplication. Then which of the following are true? (i) C∗ is cyclic (ii) Every finite subgroup of C∗ is cyclic (iii) C∗ has finitely many finite subgroups (iv) Every proper subgroup of C∗ is cyclic. 113. [Dec 2016 - C] Let R be a finite non-zero commutative ring with unity. Then which of the following statements are necessarily true? (i) Any non-zero element of R is either a unit or a zero divisor. (ii) There may exist a non-zero element of R which is neither a unit nor a zero divisor. (iii) Every prime ideal of R is maximal. (iv) If R has no zero divisors then order of any additive subgroup of R is a prime power. 114. [Dec 2016 - C] Which of the following statements are true? (i) Z[ x ] is a principal ideal domain. (ii) Z[ x, y]/ < y + 1 > is a unique factorization domain. 13  (iii)If R is a principal ideal domain and P is a non-zero prime ideal, then R/P has finitely many prime ideals. (iv) If R is a principal ideal domain, then any subring of R containing 1 is again a principal ideal domain. 115. [Dec 2016 - C] Let R be a commutative ring with unity and R[ x ] be the N polynomial ring in one variable. For a non zero f = ∑ an x n , define n =0 ω ( f ) to be the smallest n such that an 6= 0. Also ω (0) = +∞. Then which of the following statements is/are true? (i) ω ( f + g) ≥ min{ω ( f ), ω ( g)}. (ii)ω ( f g) ≥ ω ( f ) + ω ( g). (iii) ω ( f + g) = min{ω ( f ), ω ( g)}, if ω ( f ) 6= ω ( g). (iv) ω ( f g) = ω ( f ) + ω ( g), if R is an integral domain. 116. [Dec 2016 - C] Let F2 be the finite field of order 2. Then which of the following statements are true? (i) F2 [ x ] has only finitely many irreducible elements. (ii) F2 [ x ] has exactly one irreducible polynomial of degree 2. (iii) F2 [ x ]/ < x2 + 1 > is a finite dimensional vector space over F2. (iv) Any irreducible polynomial in F2 [ x ] of degree 5 has distinct roots in any algebraic closure of F2. 117. [June 2017 - B] Let S be the set of all integers from 100 to 999 which are neither divisible by 3 not divisible by 5. The number of elements is S (i) 480 (ii) 420 (iii) 360 (iv) 240. 118. [June 2017 - B] The remainder obtained when 162016 divided by 9 equals (i) 1 (ii) 2 (iii) 3 (iv) 7. 119. [June 2017 - B] Consider the ideal I = h x2 + 1, yi in the polynomial ring C[ x, y]. Which of the following statements is true? (i) I is a maximal ideal (ii) I is a prime ideal but not a prime ideal (iii) I is a maximal ideal but not a prime ideal (iv) I is neither a prime ideal nor a maximal ideal. 120. [June 2017 - C] For an integer n ≥ 2, let Sn be the permutation group on n letters and An the alternating group. Let C∗ be the group of non- zero complex numbers under multiplication. Which of the following are correct statements? (i) For every integer n ≥ 2, there is a nontrivial homomorphism χ : Sn → C∗ (ii) For every integer n ≥ 2, there is a unique nontrivial homomorphism χ : Sn → C∗ (iii) For every integer n ≥ 3, there is a nontrivial homomorphism χ : A n → C∗ (iv) For every integer n ≥ 5, there no nontrivial homomorphism χ : A n → C∗. 14 121. [June 2017 - C] Let R = { f : {1, 2, · · · , 10} → Z2 } be the set of all Z2 - valued functions on the set {1, 2, · · · , 10} of the first ten positive integers. Then R is commutative ring with pointwise addition and pointwise mul- tiplication of functions. Which of the following statements are correct? (i) R has a unique maximal ideal (ii) Every prime ideal of R is also maximal (iii) Number of proper ideals of R is 511 (iv) Every element of R is idempotent. 122. [June 2017 - C] Which of the following rings are principal ideal do- mains(PID)? (i) Q[ x ] (ii) Z[ x ] (iii) (Z/6Z[ x ])[ x ] (iv) (Z/7Z[ x ])[ x ] 123. [June 2017 - C] Let G be a group of order 125. Which of the following statements are necessarily true? (i) G has a non-trivial abelian subgroup (ii) The centre of G is a proper subgroup (iii) The centre of G has a order 5 (iv) There is a subgroup of order 25. 124. [June 2017 - C] Let R be a non-zero ring with identity such that a2 = a. Which of the following statements are true? (i) There is no such ring (ii) 2a = 0 for all a ∈ R (iii) 3a = 0 for all a ∈ R (iv) Z/2Z is a subring of R. 125. [June 2017 -C] Which of the following polynomials are irreducible in Z[ x ] ? (i) x4 + 10x + 5 (ii) x3 − 2x + 1 (iii) x4 + x2 + 1 (iv) x3 + x + 1. 126. [Dec 2017 -B]Let f : Z → (Z/4Z) × (Z/6Z) be the function f (n) = (n mod 4, n mod 6). Then (i) (0 mod 4, 3 mod 6) is in the image of f (ii) ( a mod 4, b mod 6) is in the image of f , for all even integers a and b (iii) image of f has exactly 6 elements (iv) kernal of f = 24Z. 127. [Dec 2017 -B] The group S3 of permutations of {1, 2, 3} acts on the three dimensional vector space over the finite field F3 of three elements, by permuting the vectors in basis {e1 , e2 , e3 } by σ · e1 = eσ(1) , for all σ ∈ S3. The cardinality of the set of vectors fixed under the above action is (i) 0 (ii) 3 (iii) 9 (iv)27. 128. [Dec 2017 -B] Let R be a subring of Q containing 1. Then which of the following is necessarily true? (i) R is a principal ideal domain (PID) (ii) R contains infinitely many prime ideals 15  (iii) R contains a prime ideal which is not a maximal ideal (iv) for every maximal ideal M in R, the residue field R/M is finite. 129. [Dec 2017 - C]Let G be a finite abelian group and a, b ∈ G with order ( a) = m, order (b) = n. Which of the following are necessarily true? (i) order ( ab) = mn (ii) order ( ab) = lcm(m, n) (iii) there is an element of G whose order is lcm(m, n) (iv) order ( ab) = gcd(m, n). 130. [Dec 2017 -C] Which of the following rings are principal ideal domains (PIDs)? (i) Z[ X ]/h X 2 + 1i (ii) Z[ X ] (iii) C[ X, Y ] (iv) R[ X, Y ]/h X 2 + 1, Y i. 131. [Dec 2017 -C] For any prime number p, let A p be the set of integers d ∈ {1, 2, · · · , 999} such that the power of p in the prime factorisation of d is odd. Then the cardinality of (i) A3 is 250 (ii) A5 is 160 (iii) A7 is 124 (iv) A11 is 82. 132. [Dec 2017 -C] Let F be a finite field and let K/F be a field extension of degree 6. Then the Galois group of K/F is isomorphic to (i) the cyclic group of order 6 (ii) the permutation group on {1, 2, 3} (iii) the permutation group on {1, 2, 3, 4, 5, 6} (iv) the permutation group on {1}. 16