CSIR  Algebra Problems


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1 CSIR  Algebra Problems N. Annamalai DST  INSPIRE Fellow (SRF) Department of Mathematics Bharathidasan University Tiruchirappalli algebra.annamalai@gmail.com Website: 1. [June B] Let α = (1, 3, 5, 7, 9, 11) and β = (2, 4, 6, 8) be two permutations in S 100. Then the order of αβ is (i) 4 (ii) 6 (iii) 12 (iv) [June B] The number of groups of order 121, up to isomorphism, is (i) 1 (ii) 2 (iii) 11 (iv) [June B] Let G a nonabelian group of order 21. Let H be a Sylow 3subgroup and K be a Sylow 7subgroup 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. 4. [June B] Let f(x) Z 5 [x] be a polynomial such that field. Then one of the choices for f(x) is (i) x (ii) x (iii) x (iv) x Z 5[x] <f(x)> is a 5. [Dec 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 IJ = {xy x I, y J}. Then (i) I J is an ideal and IJ is an ideal in R (ii) I J is an ideal and IJ need not be an ideal in R (iii) I J need not be an ideal but IJ is an ideal in R (iv) Neither I J nor IJ need to be an ideal in R. 6. [Dec B] The polynomial x 3 + 5x is (i) irreducible over Z but reducible over Z 5 (ii) irreducible over both Z and Z 5 (iii) reducible over Z but irreducible over Z 5 (iv) reducible over both Z and Z [Dec B] In Z[i] (i) 5 and 6 are irreducible 1
2 N. Annamalai 2 (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 B] The number of distinct homomorphisms from Z 12 to Z 25 is (i) 1 (ii) 2 (iii) 3 (iv) [Dec B] The number of 5Sylow subgroups of S 6 is (i) 16 (ii) 6 (iii) 36 (iv) [Dec 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) [June B] The number of generators of a cyclic group of order 12 is (i) 1 (ii) 2 (iii) 3 (iv) [June B] Up to an isomorphism, the number of groups of order 33 is (i) 3 (ii) 11 (iii) 1 (iv) Infinitely many. 13. [June B] Let R = Q[x]. Let I be the principal ideal < x > and J be the principal ideal < x 2 >. 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 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 B] The polynomial x 3 7x x 9 is (i) irreducible over both Z and Z 3 (ii) irreducible over Z but reducible over Z 3 (iii) reducible over Z but irreducible over Z 3 (iv) reducible over both Z and Z [June B] A permutation a of {1, 2,, n} is called a derangement if α(i) i for every i. Let d n denote the number of derangements of {1, 2,, n}. Then d 4 is equal to (i) 3 (ii) 9 (iii) 12 (iv) [June B] The number of sub fields of a finite field of order 3 10 is equal to (i) 4 (ii) 5 (iii) 3 (iv) [June B] The unit digit of is (i) 2 (ii) 4 (iii) 6 (iv) [June B] The degree of the extension Q( ) over the field Q( 2) is (i) 1 (ii) 2 (iii) 3 (iv) 6.
3 N. Annamalai [June B] Consider a group G. Let Z(G) be its centre. For n N, define J n = {(g 1, g 2,, g n ) Z(G) Z(G) Z(G) : g 1 g 2 g n = e}. As a subset of a direct product groups G G (n times direct product of the group G), J n 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 B] Let I 1 be the ideal generated by x 4 + 3x and I 2 be the ideal generated by x in Q[x]. If F 1 = Q[x]/I 1 and F 2 = Q[x]/I 2, then (i) F 1 and F 2 are fields (ii) F 1 is a field but F 2 is not a field (iii) F 1 is not a field while F 2 is a field (iv) neither F 1 nor F 2 is a field 22. [June B] Let G be group of order 77. Then the centre of G is isomorphic to (i) Z 1 (ii) Z 7 (iii) Z 11 (iv) Z [June C] Let G = Z 10 Z 15. 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 [June 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 S 4. Then (i) H and K are normal subgroups of S 4 (ii) H is normal in K and K is normal in A 4 (iii) H is normal in A 4 but not normal in S 4 (iv) K is normal in S 4 but H is not. 25. [June C] Let < p(x) > denote the ideal generated by the polynomial p(x) in Q[x]. If f(x) = x 3 + x 2 + x + 1 and g(x) = x 3 x 2 + x 1, then (i) < f(x) > + < g(x) >=< x 3 + x > (ii) < f(x) > + < g(x) >=< f(x)g(x) > (iii) < f(x) > + < g(x) >=< x > (iv) < f(x) > + < g(x) >=< x 4 1 > 26. [June C] Let I 1 be the ideal generated by x and I 2 be the ideal generated by x 3 x 2 + x 1 in Q[x]. If R 1 = Q[x]/I 1 and R 2 = Q[x]/I 2, then (i) R 1 and R 2 are fields (ii) R 1 is a field but R 2 is not a field (iii) R 1 is an integral domain but R 2 is not an integral domain (iv) R 1 and R 2 are not integral domains. 27. [Dec B] Let p be a prime number. The order of a psylow subgroup of the group GL 50 (F p ) of invertible matrices with entries from the finite fieldf p, equals: (i) p 50 (ii) p 125 (iii) p 1250 (iv) p 1225
4 N. Annamalai [Dec B] The number of multiples of that divide is (i) 11 (ii) 12 (iii) 121 (iv) [Dec B] The number of group homomorphisms from the symmetric group S 6 to Z 6 is (i) 1 (ii)2 (iii)3 (iv) [Dec 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) [Dec C] Which of the following integral domains are Euclidean domains? (i) Z[ ) 3] (ii) Z[x] (iii) R[x 2, x 3 ] (iv) [y]. ( Z[x] (2,x) 32. [Dec C] Let G be the Galois group of the splitting field of x 5 2 over Q. Then, which of the following statements are true? (i) G is cyclic (ii) G is nonabelian (iii) the order of G is 20 (iv) G has an element of order [June B] The number of positive divisors of 50,000 is (i) 20 (ii) 30 (iii) 40 (iv) [June B] The number 2e ix is (i) a rational number (ii) a transcendental number (iii) an irrational number (iv) an imaginary number. 35. [June B] Let F be a field of 8 elements and A = {x F : x 7 = 1 and x k 1 for all natural number k < 7}. Then the number of elements of A is (i) 1 (ii) 2 (iii) 3 (iv) [June 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 B] Let f(x) = x 3 + 2x and g(x) = 2x 2 + x + 2. Then over Z 3 (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 B] The number of nontrivial ring homomorphisms from Z 12 to Z 28 is (i) 1 (ii) 3 (iii) 4 (iv) [June C] Let R = Q/I where I is the ideal generated by 1 + x 2. Let y be a coset of x in R. Then (i) 1 + y 2 is irreducible over R (ii) 1 + y + y 2 is irreducible over R (iii) y 2 y + 1 is irreducible over R (iv) 1 + y + y 2 + y 3 is irreducible over R.
5 N. Annamalai [June C] which of the following is true? (i) sin 7 o is algebraic over Q (ii) cos π/17 is algebraic over Q (iii) sin 1 1 is algebraic over Q (iv) 2 + π is algebraic over Q()π. 41. [June C] Let f(x) = x 3 + x 2 + x + 1 and g(x) = x Then in Q[x], (i) gcd(f(x), g(x)) = x + 1 (ii) gcd(f(x), g(x)) = x 2 1 (iii) lcm(f(x), g(x)) = x 5 + x 3 + x (iv) lcm(f(x), g(x)) = x 5 + x 4 + x 3 + x [June 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 C] Let G denote the group S 4 S 3. Then (i) a 2sylow subgroup of G is normal (ii) a 3sylow subgroup of G is normal (iii) G has a nontrivial normal subgroup (iv) G has a normal subgroup of order [Dec B] The last two digits of 7 81 are (i) 07 (ii) 17 (iii) 37 (iv) [Dec B] In which of the following fields, the polynomial x x is irreducible in F[x]? (i) the field F 3 with 3 elements (ii) the field F 7 with 7 elements (iii) the field F 13 with 13 elements (iv) the field Q of rational numbers. 46. [Dec B] let ω be a complex number such that ω 3 = 1 and ω 1. Suppose L is the field Q( 3 2, ω) generated by 3 2 and ω over the field 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) [Dec 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(a n 1) for every positive integers a and n (iii)n divides ϕ(a n 1) for every positive integers a and n such that GCD(a, n) = 1 (iv) a divides ϕ(a n 1) for every positive integers a and n such that GCD(a, n) = [Dec C] For a positive integer n 4 and a prime number p n, U p,n denote the union of all psylow subgroups of a alternating group A n on n letters. Also let K p,n denote a subgroup of A n generated by U p,n, and let K p,n denote the order of K p,n. Then (i) K 2,4 = 12 (ii) K 2,4 = 4 (iii) K 2,5 = 60 (iv) K 3,5 = 30.
6 N. Annamalai [Dec 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 p e(x) is an irreducible polynomial in Q[x] for every prime number p and for every positive integer e. (iv) f p (x pe 1 ) is an irreducible polynomial in Q[x] for every prime number p and for every positive integer e. 50. [Dec C] Consider the ring R = Z[ 5] = {a + b 5 : a, b Z} and the element α = 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 C] Consider the polynomial f(x) = x 4 x x 2 + 5x 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 F 3, it has no roots in F 3. (ii) Considering f as a polynomial with coefficients in F 3, it is a product of two irreducible factors of degree 2 over F 3. (iii) Considering f as a polynomial with coefficients in F 7, it has an irreducible factor of degree 3 over F 7. (iv) f is a product of two polynomials of degree 2 over Z. 52. [Dec C] For a positive integer m, let a m denote the number of Q[x] distinct prime ideals of the ring <x m 1>. Then (i) a 4 = 2 (ii) a 4 = 3 (iii) a 5 = 2 (iv) a 5 = [June 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) [June B] What is the smallest positive integer in this set {24x + 60y z x, y, z Z}? (i) 2 (ii) 4 (iii) 6 (iv) [June B] Which of the following ring is PID? (i) Q[x, y]/ < x > (ii) Z Z (iii) Z[x] (iv) M 2 (Z) the ring of all 2 2 matrices with entries in Z. 56. [June B] Let F C be the splitting field of x 7 2 over Q, and z = e 2πi/7, a primitive seventh root of unity. Let [F : Q(z)] = a and [F : Q( 7 2)] = b. Then (i) a = b = 7 (ii) a = b = 6 (iii) a > b (iv) a < b. 57. [June C] Let σ = (12)(345) and τ = (123456) be permutations in S 6. 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 S 6 (iii) < σ > < τ > is a trivial group (iv) σ and τ commute.
7 N. Annamalai [June C] Let S n denote the symmetric group on n symbols. The group S 3 (Z/2Z) is isomorphic to which of the following groups? (i) Z/12Z (ii) Z/6Z Z/2Z (iii) A 4, the alternating group of order 12 (iv) D 6, the dihedral group of order [June C] Let F = F 3 [x]/(x 3 + 2x 1), where F 3 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 F 3 (iii) The automorphism group of F is cyclic (iv) The automorphism group of F is abelian but not cyclic. 60. [June C] Which of the following polynomials are irreducible over the given rings? (i) x 5 + 3x 4 + 9x + 15 over Q (ii) x 3 + 2x 2 + x + 1 over Z 7 (iii) x 3 + x 2 + x + 1 over Z (iv) x 4 + x 3 + x 2 + x + 1 over Z. 61. [Dec B] How many normal subgroups does a nonabelian group G of order 21 have other than {e} and G? (i) 0 (ii) 1 (iii) 3 (iv) [Dec B] The number of group homomorphisms from the symmetric group S 3 to the additive group Z 6 is (i) 1 (ii) 2 (iii) 3 (iv) [Dec C] Determine which of the following cannot be the class equation of a group (i) 10 = (ii) 4 = (iii) 8 = (iv) 6 = [Dec C] Let F and F be two finite fields order q and q respectively. Then (i) F contains a subfield isomorphic to F if and only if q q (ii) F contains a subfield isomorphic to F if and only if q divides q (iii) If the gcd of q and q is not 1, then both are isomorphic to subfields of some finite field L (iv) Both F and F are quotient rings of the ring Z[x]. 65. [Dec C] Let R be a nonzero commutative ring with unity 1 R. Define the characteristic of R to be the order 1 R in (R, +) if it is finite and to be zero if the order of 1 R in (R, +) is infinite. We denote the characteristic of R by char(r). In the following, let R and S be nonzero 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
8 N. Annamalai 8 (iii) if S is a subring of R containing 1 R then char(s) = char(r) (iv) if char(r) is a prime number, then R is a field. 66. [Dec 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 C] Let f(x) = x 3 + 2x 2 + 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 = F 2, the finite field of 2 elements (iv) K = F 3, the finite field of 3 elements. 68. [June B] The total number of nonisomorphic groups of order 121 is (i) 2 (ii) 1 (iii) 61 (iv) [June B] Let G denote the group of all automorphisms of the field F that consists of elements. Then the number of distinct subgroups of G is equal to (i) 4 (ii) 3 (iii) 100 (iv) [June B] Let p, q be distinct primes. Then (i) Z/p 2 qz has exactly 3 distinct ideals (ii) Z/p 2 qz has exactly 3 distinct prime ideals (iii) Z/p 2 qz has exactly 2 distinct prime ideals (iv) Z/p 2 qz has a unique maximal ideal. 71. [June C] Let f(x) = x 4 +3x 3 9x 2 +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 Z 2 (ii) f(x) is irreducible over Q (iii) f 3 (x) is irreducible over Z 3 (iv) f(x) is irreducible over Z. 72. [June 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 C] Consider the multiplicative group G of all the (complex) 2 n 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.
9 N. Annamalai [June 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 C] Pick the correct statements. (i) Q( 2) and Q(i) are isomorphic as Qvector spaces (ii) Q( 2) and Q(i) are isomorphic as fields (iii) Gal Q (Q( 2)/Q) = Gal Q (Q(i)/Q) (iv) Q( 2) and Q(i) are both Galois extensions of Q. 76. [Dec B] Find the degree of the field extension Q( 2, 4 2, 8 2) over Q. (i) 4 (ii) 8 (iii) 14 (iv) [Dec 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) [Dec B] The number of conjugacy classes in the permutation group S 6 is (i) 12 (ii) 11 (iii) 10 (iv) [Dec B] In the group of all invertible 4 4 matrices with entries in the field of 3 elements, any 3sylow subgroup has cardinality (i) 3 (ii) 81 (iii) 243 (iv) [Dec C] Let G be a nonabelian group. Then, its order can be: (i) 25 (ii) 55 (iii) 125 (iv) [Dec 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 anonzero prime ideal (ii) I is a maximal ideal iff the quotient ring R[x]/I is isomorphic to R (iii) I is a maximal ideal iff I = (f(x)) where f(x) is a nonconstant irreducible polynomial over R (iv) I is a maximal ideal iff there exist a nonconstant polynomial f(x) I of degree [Dec 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 [Dec 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
10 N. Annamalai 10 (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 B] The number of subfields of a field of cardinality is (i)2 (ii) 4 (iii) 9 (iv) [June B] Up to isomorphism, the number of abelian groups of order 108 is (i) 12 (ii) 9 (iii) 6 (iv) [June B] Let R be the ring Z[x]/((x 2 + x + 1)(x 3 + 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 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) k} has an even number of elements (iv) The set {k σ(k) = k} has an odd number of elements 88. [June 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 = [June C] Which of the following cannot be the class equation of a group of orde 10? (i) = 10 (ii) = 10 (iii) = 10 (iv) = [June 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 (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 C] Determine Which of the following polynomials are irreducible over the indicated rings. (i) x 5 3x 4 + 2x 3 5x + 8 over R (ii) x 3 + 2x 2 + x + 1 over Q (iii) x 3 + 3x 2 6x + 3 over Z (iv) x 4 + x over Z/2Z. 92. [Dec B] A group G is generated by the elements x, y with the relations x 3 = y 2 = (xy) 2 = 1. The order of G is (i)4 (ii) 6 (iii) 8 (iv) 12.
11 N. Annamalai [Dec 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 B] Which of the following is an irreducible factor of x 12 1 over Q? (i) x 8 + x (ii) x (iii) x 4 x (iv) x 5 x 4 + x 3 x 2 + x [Dec C] Let a n denote the number of those permutations σ on {1, 2,, n} such that σ is a product of exactly two disjoint cycles. Then: (i) a 5 = 50 (ii) a 4 = 14 (iii) a 5 = 40 (iv) a 4 = [Dec C] Let G be a simple group of order 60. Then (i) G has six Sylow5 subgroups (ii) G has four Sylow3 subgroups. (iii) G has a cyclic subgroup of order 6. (iv) G has a unique element of order [Dec C] Let A denote the quotient ring Q[x]/(x 3 ). 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) = [Dec C] Which of the following quotient rings are fields? (i) F 3 [x]/(x 2 + x + 1), where F 3 is the finite field with 3 elements. (ii) Z[x]/(x 3) (iii) Q[x]/(x 2 + x + 1) (iv) F 2 [x]/(x 2 + x + 1), where F 2 is the finite field with 2 elements. 99. [Dec C] Let ω = cos 2π 2π 10 +i sin 10. Let K = Q(ω2 ) and let L = Q(ω). Then (i) [L : Q] = 10 (ii) [L : K] = 2 (iii) [K : Q] = 4 (iv) L = K [Dec 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/2 3 Z) (iii) (Z/100Z) (iv) (Z/163Z) [June B] Let p be a prime number. How many distinct subrings (with unity) of cardinality p does the field F p 2 have? (i) 0 (ii) 1 (iii) p (iv) p [June 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.
12 N. Annamalai [June 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 H 1 H 2 where H 1, H 2 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 [June 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 [June C] Consider the symmetric group S 20 and its subgroup A 20 consisting of all even permutations. Let H be a 7Sylow subgroup of A 20. Pick each correct statement from below: (i) H = 49 (ii) H must be cyclic (iii) H is a normal subgroup of A 20 (iv) Any 7Sylow subgroup of S 20 is a subset of A [June C] Let pbe a prime. Pick each correct statement from below. Up to isomorphism, (i) there are exactly two abelian groups of order p 2 (ii) there are exactly two groups of order p 2 (iii) there are exactly two commutative rings of order p 2 (iv) there is exactly one integral domain of order p [June 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 [June 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 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]. f(x) 109. [Dec B] Let S n denote the permutation group on n symbols and A n be the subgroup of even permutations. Which of the following is true? (i) There exists a finite group which is not a subgroup of S n for any n 1. (ii) Every finite group is a subgroup of A n for some n 1. (iii) Every finite group is a quotient of A n for some n 1. (iv) No finite abelian group is a quotient of S n for n > 3.
13 N. Annamalai [Dec B] What is the number of nonsingular 3 3 matrices over F 2, the finite field with two elements? (i) 168 (ii) 384 (iii) 2 3 (iv) [Dec C] Consider the following subsets of the group of 2 2 nonsingular {( matrices ) over R : } {( ) } 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 welldefined 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 [Dec 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 [Dec C] Let R be a finite nonzero commutative ring with unity. Then which of the following statements are necessarily true? (i) Any nonzero element of R is either a unit or a zero divisor. (ii) There may exist a nonzero 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 [Dec 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. (iii)if R is a principal ideal domain and P is a nonzero 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 [Dec C] Let R be a commutative ring with unity and R[x] be the polynomial ring in one variable. For a non zero f = N a n x n, define ω(f) to be the smallest n such that a n 0. Also ω(0) = +. Then which of the following statements is/are true? (i) ω(f + g) min{ω(f), ω(g)}. (ii)ω(fg) ω(f) + ω(g). (iii) ω(f + g) = min{ω(f), ω(g)}, if ω(f) ω(g). (iv) ω(fg) = ω(f) + ω(g), if R is an integral domain. n=0
14 N. Annamalai [Dec C] Let F 2 be the finite field of order 2. Then which of the following statements are true? (i) F 2 [x] has only finitely many irreducible elements. (ii) F 2 [x] has exactly one irreducible polynomial of degree 2. (iii) F 2 [x]/ < x > is a finite dimensional vector space over F 2. (iv) Any irreducible polynomial in F 2 [x] of degree 5 has distinct roots in any algebraic closure of F 2.
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