Comprehensive Multiple-Choice Question Set: Algebraic Structures -
Groups, Rings, and Fields
1. What is the order of the symmetric group S ?₃
a) 3
b) 6
c) 9
d) 12
Answer: b) 6
2. Which of the following is not a group?
a) (Z, +)
b) (Q*, ×)
c) (R, +)
d) (N, ×)
Answer: d) (N, ×)
3. What is the identity element in the group (R*, ×)?
a) 0
b) 1
c) -1
d) ∞
Answer: b) 1
4. In a group G, if a² = e for all a G, what type of group is G?∈
a) Cyclic
b) Abelian
c) Simple
d) Nilpotent
Answer: b) Abelian
5. What is the order of an element a in a finite group if a⁵ = e?
a) 5
b) A factor of 5
c) A multiple of 5
d) Cannot be determined
Answer: b) A factor of 5
6. Which of the following is an example of a ring with unity?
a) (Z, +, ×)
b) (2Z, +, ×)
c) (Q, +, -)
d) (R, +, ÷)
Answer: a) (Z, +, ×)
7. What is a field?
a) A commutative ring with unity where every non-zero element has a multiplicative inverse
b) A group with two binary operations
c) A ring without zero divisors
d) An abelian group under addition
Answer: a) A commutative ring with unity where every non-zero element has a multiplicative
inverse
8. Which of the following is not a subgroup of (R, +)?
a) (Z, +)
b) (Q, +)
c) ({0}, +)
d) (R*, ×)
Answer: d) (R*, ×)
9. What is the order of the alternating group A ?₄
a) 4
b) 8
c) 12
d) 24
Answer: c) 12
10. In the ring of integers modulo 6, Z , what is 4 + 5?₆
a) 3
b) 9
c) 0
d) 1
Answer: a) 3
11. Which of the following is an integral domain but not a field?
a) Z
b) Q
c) R
d) C
Answer: a) Z
12. What is the characteristic of the field of real numbers R?
a) 0
b) 1
c) 2
d) ∞
Answer: a) 0
13. If G is a group and H is a subgroup of G, what is [G:H] called?
a) Order of H
b) Index of H in G
c) Degree of H
d) Rank of H
Answer: b) Index of H in G
14. Which of the following is not a property of homomorphisms?
a) f(ab) = f(a)f(b)
b) f(e) = e'
c) f(a ¹) = [f(a)] ¹⁻ ⁻
d) f is always bijective
Answer: d) f is always bijective
15. What is the center of a group G?
a) The set of all elements that commute with every element in G
b) The set of all subgroups of G
c) The set of all generators of G
d) The set of all invertible elements in G
Answer: a) The set of all elements that commute with every element in G
16. Which of the following is true for any field F?
a) F is a ring
b) F has no zero divisors
c) F is commutative under multiplication
d) All of the above
Answer: d) All of the above
17. What is the order of the quaternion group Q ?₈
a) 4
b) 6
c) 8
d) 16
Answer: c) 8
18. In a ring R, an element a is called nilpotent if:
a) a² = a
b) aⁿ = 0 for some positive integer n
c) ab = 0 for some non-zero b in R
d) a has no multiplicative inverse
Answer: b) aⁿ = 0 for some positive integer n
19. Which of the following is not a field?
a) Q
b) R
c) C
d) Z
Answer: d) Z
20. What is the multiplicative inverse of 3 in Z ?₇
a) 3
b) 4
c) 5
d) 6
Answer: c) 5
21. If G is a group of order 15, what can be said about G?
a) G is cyclic
b) G is abelian
c) G is simple
d) G is perfect
Answer: a) G is cyclic
22. Which of the following is an example of a non-commutative ring?
a) Z[x]
b) M (R) (2x2 matrices over R)₂
c) Z[i]
d) Q[x]
Answer: b) M (R) (2x2 matrices over R)₂
23. What is the kernel of a group homomorphism f: G → H?
a) {g G | f(g) = e_H}∈
b) {h H | f ¹(h) = e_G}∈ ⁻
c) {g G | f(g) = g}∈
d) {h H | f(h) = h}∈
Answer: a) {g G | f(g) = e_H}∈
24. Which of the following is true for any ideal I of a ring R?
a) I is a subring of R
b) I is closed under addition
c) For all r R and a I, ra and ar are in I∈ ∈
d) All of the above
Answer: d) All of the above
25. What is the order of the multiplicative group of units in Z ?₁₂
a) 2
b) 3
c) 4
d) 6
Answer: c) 4
26. Which of the following is not a property of fields?
a) Associativity of addition and multiplication
b) Commutativity of addition and multiplication
c) Distributivity of multiplication over addition
d) Existence of additive inverses for all elements
Answer: d) Existence of additive inverses for all elements (this is true, but not a distinguishing
property of fields)
27. What is the Galois group of a splitting field of x² - 2 over Q?
a) Trivial group
b) Z₂
c) S₂
d) D₄
Answer: b) Z₂
28. In a ring R, if ab = 0 implies a = 0 or b = 0, what type of ring is R?
a) Commutative ring
b) Ring with unity
c) Integral domain
d) Field
Answer: c) Integral domain
29. What is the characteristic of the field Z ?₅
a) 0
b) 1
c) 5
d) ∞
Answer: c) 5
30. Which of the following is an automorphism of the group (Z, +)?
a) f(x) = x + 1
b) f(x) = -x
c) f(x) = 2x
d) f(x) = x²
Answer: b) f(x) = -x
31. What is the quotient ring Z[x]/(x² + 1) isomorphic to?
a) Z
b) Z[i]
c) Q
d) R
Answer: b) Z[i]
32. If G is a group and H is a normal subgroup of G, what is G/H?
a) A subgroup of G
b) A quotient group
c) A ring
d) A field
Answer: b) A quotient group
33. Which of the following is true for a finite field F?
a) |F| = pⁿ for some prime p and positive integer n
b) F is always commutative
c) F has a multiplicative group of order |F| - 1
d) All of the above
Answer: d) All of the above
34. What is the degree of the minimal polynomial of √2 over Q?
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
35. In a group G, if a⁴ = e and a² ≠ e, what is the order of a?
a) 2
b) 3
c) 4
d) 8
Answer: c) 4
36. Which of the following is not a property of ring homomorphisms?
a) f(a + b) = f(a) + f(b)
b) f(ab) = f(a)f(b)
c) f(1) = 1 if the rings have unity
d) f is always surjective
Answer: d) f is always surjective
37. What is the multiplicative group of the field F ?₄
a) Z₂
b) Z₃
c) Z₄
d) S₃
Answer: b) Z₃
38. If R is a ring and I is an ideal of R, what is R/I?
a) A subring of R
b) A quotient ring
c) A field
d) A group
Answer: b) A quotient ring
39. Which of the following is an example of a principal ideal domain that is not a field?
a) Z
b) Q[x]
c) R[x, y]
d) M (R)₂
Answer: a) Z
40. What is the order of GL (Z ), the group of invertible 2x2 matrices over Z ?₂ ₂ ₂
a) 4
b) 6
c) 8
d) 16
Answer: b) 6
41. In a field F, what is the multiplicative inverse of 0?
a) 1
b) -1
c) 0
d) Does not exist
Answer: d) Does not exist
42. Which of the following is not a subfield of C?
a) Q
b) R
c) Q(i)
d) Z
Answer: d) Z
43. What is the Sylow 2-subgroup of S ?₄
a) Z₂
b) Z × Z₂ ₂
c) D₄
d) Q₈
Answer: c) D₄
44. In the ring Z , what are the zero divisors?₆
a) {0, 2, 3, 4}
b) {0, 2, 4}
c) {0, 3}
d) {2, 3, 4}
Answer: c) {0, 2, 3, 4}
45. What is the splitting field of x³ - 2 over Q?
a) Q( 2)∛
b) Q( 2, ω) where ω is a primitive cube root of unity∛
c) Q(i)
d) R
Answer: b) Q( 2, ω) where ω is a primitive cube root of unity∛
46. Which of the following is an example of a non-abelian simple group?
a) Z₅
b) Q₈
c) A₅
d) D₆
Answer: c) A₅
47. What is the order of the automorphism group of Z ?₄
a) 1
b) 2
c) 3
d) 4
Answer: b) 2
48. In a field F, what is the characteristic polynomial of the linear transformation T(x) = 2x?
a) x - 2
b) x + 2
c) x² - 2
d) x² - 4
Answer: a) x - 2
49. Which of the following is not a maximal ideal in Z?
a) (2)
b) (3)
c) (4)
d) (5)
Answer: c) (4)
50. What is the degree of the extension Q(√2, √3) over Q?
a) 2
b) 3
c) 4
d) 6
Answer: c) 4
51. In a group G, if |G| = pq where p and q are distinct primes, what can be said about G?
a) G is always cyclic
b) G is always abelian
c) G is simple if and only if it is non-abelian
d) G has a normal subgroup of order p or q
Answer: d) G has a normal subgroup of order p or q
52. Which of the following is an example of a Euclidean domain?
a) Z[x]
b) Z[i]
c) Z[√-5]
d) Q[x, y]
Answer: b) Z[i]
53. What is the centralizer of an element a in a group G?
a) {g G | ga = ag}∈
b) {g G | gag ¹ = a}∈ ⁻
c) {g G | g ¹ag = a}∈ ⁻
d) All of the above
Answer: d) All of the above
54. In a field F, what is the minimal polynomial of a root of x⁴ + 1?
a) x² + 1
b) x² - 1
c) x⁴ + 1
d) Cannot be determined without more information
Answer: d) Cannot be determined without more information
55. What is the Galois group of x³ - 2 over Q?
a) Z₂
b) Z₃
c) S₃
d) A₃
Answer: c) S₃
56. Which of the following is true for any finite group G?
a) |G| = lcm{|a| : a G}∈
b) G has an element of order 2 if |G| is even
c) G is cyclic if and only if it has a unique subgroup of each order dividing |G|
d) G is always solvable
Answer: b) G has an element of order 2 if |G| is even
57. What is the dimension of Q(√2, √3, √5) as a vector space over Q?
a) 3
b) 4
c) 6
d) 8
Answer: d) 8
58. In a ring R, what is the annihilator of an element a?
a) {r R | ra = 0}∈
b) {r R | ar = 0}∈
c) {r R | ra = ar = 0}∈
d) {r R | ra = a}∈
Answer: c) {r R | ra = ar = 0}∈
59. Which of the following is not a property of simple groups?
a) They have no non-trivial normal subgroupsC
60. Which of the following is not a property of simple groups?
a) They have no non-trivial normal subgroups
b) They are always abelian
c) Their order is always prime or composite
d) They are building blocks for all finite groups
Answer: b) They are always abelian
61. What is the multiplicative group of the field F ?₉
a) Z₈
b) Z₆
c) Z × Z₃ ₃
d) S₃
Answer: a) Z₈
62. In a ring R, what is a unit?
a) An element with a multiplicative inverse
b) An element with an additive inverse
c) An element that commutes with all other elements
d) An element that divides 1
Answer: a) An element with a multiplicative inverse
63. What is the Galois group of x⁴ - 1 over Q?
a) Z₂
b) Z₄
c) Z × Z₂ ₂
d) D₄
Answer: c) Z × Z₂ ₂
64. Which of the following is an example of a non-commutative division ring?
a) Q
b) R
c) C
d) Quaternions
Answer: d) Quaternions
65. What is the order of the group of automorphisms of Z ?₆
a) 1
b) 2
c) 3
d) 6
Answer: b) 2
66. In a field F, what is the minimal polynomial of a primitive 5th root of unity?
a) x - 1
b) x² + x + 1
c) x⁴ + x³ + x² + x + 1
d) x⁵ - 1
Answer: c) x⁴ + x³ + x² + x + 1
67. Which of the following is true for any finite field F?
a) |F| = p for some prime p
b) F is always an extension of Z₂
c) The multiplicative group F* is cyclic
d) F has characteristic 0
Answer: c) The multiplicative group F* is cyclic
68. What is the rank of the free abelian group Z × Z × Z?
a) 1
b) 2
c) 3
d) Infinite
Answer: c) 3
69. In a ring R, what is a prime ideal?
a) An ideal that contains only prime elements
b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
c) An ideal that is contained in every maximal ideal
d) An ideal generated by a single element
Answer: b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
70. What is the degree of the splitting field of x³ - 2 over Q?
a) 3
b) 4
c) 6
d) 8
Answer: c) 6
71. Which of the following is not a subring of Q[x]?
a) Z[x]
b) Q
c) Z
d) Q[x²]
Answer: b) Q
72. What is the center of the dihedral group D ?₈
a) {e}
b) {e, r²}
c) {e, r, r², r³}
d) D₈
Answer: b) {e, r²}
73. In a field F, what is the minimal polynomial of √2 + √3?
a) x² - 5
b) x² - 2x - 1
c) x⁴ - 10x² + 1
d) x⁴ - 5x² + 6
Answer: c) x⁴ - 10x² + 1
74. Which of the following is an example of a local ring?
a) Z
b) Q
c) Z/(p²) where p is prime
d) R[x]
Answer: c) Z/(p²) where p is prime
75. What is the order of GL (Z ), the group of invertible 3x3 matrices over Z ?₃ ₂ ₂
a) 24
b) 168
c) 336
d) 512
Answer: b) 168
76. In a group G, what is the normalizer of a subgroup H?
a) {g G | ghg ¹ H for all h H}∈ ⁻ ∈ ∈
b) {g G | gH = Hg}∈
c) {g G | gHg ¹ = H}∈ ⁻
d) All of the above
Answer: d) All of the above
77. What is the Krull dimension of the ring Z?
a) 0
b) 1
c) 2
d) Infinite
Answer: b) 1
78. Which of the following is true for any field extension E/F?
a) [E:F] = 1 if and only if E = F
b) If [E:F] is prime, then E/F is a simple extension
c) If [E:F] is finite, then E is algebraic over F
d) All of the above
Answer: d) All of the above
79. What is the order of the group Aut(Z ), the automorphism group of Z ?₈ ₈
a) 2
b) 4
c) 8
d) 16
Answer: b) 4
80. In a ring R, what is a maximal ideal?
a) An ideal that is not contained in any other proper ideal
b) An ideal that contains all other ideals
c) An ideal generated by a maximal element
d) An ideal with the largest number of elements
Answer: a) An ideal that is not contained in any other proper ideal
81. What is the Galois group of x⁴ + 1 over Q?
a) Z₄
b) Z × Z₂ ₂
c) D₄
d) S₄
Answer: c) D₄
82. Which of the following is an example of a non-Noetherian ring?
a) Z
b) Q[x]
c) R[[x]] (formal power series over R)
d) M (R) (2x2 matrices over R)₂
Answer: c) R[[x]] (formal power series over R)
83. What is the degree of the minimal polynomial of a primitive 7th root of unity over Q?
a) 3
b) 6
c) 7
d) 14
Answer: b) 6
84. In a group G, what is the commutator subgroup?
a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
b) The largest normal subgroup of G
c) The subgroup of all elements that commute with every element in G
d) The subgroup of all elements of order 2
Answer: a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
85. Which of the following is true for any finite-dimensional vector space V over a field F?
a) V is isomorphic to F^n for some n
b) V has a unique basis
c) All subspaces of V are finite-dimensional
d) The dual space V* is isomorphic to V
Answer: a) V is isomorphic to F^n for some n
86. What is the characteristic of the field Q(i)?
a) 0
b) 2
c) 4
d) Infinite
Answer: a) 0
87. In a ring R, what is a nilpotent element?
a) An element a such that a² = 0
b) An element a such that aⁿ = 0 for some positive integer n
c) An element a such that a² = a
d) An element a such that ab = 0 for some non-zero b in R
Answer: b) An element a such that aⁿ = 0 for some positive integer n
88. What is the Sylow 3-subgroup of A ?₄
a) Z₃
b) Z × Z₃ ₃
c) A₃
d) S₃
Answer: a) Z₃
89. Which of the following is an example of a field that is not algebraically closed?
a) C
b) R
c) Q
[ (algebraic closure of Q)
d) F
[ (algebraic closure of F )₂ ₂
Answer: b) R
90. What is the order of the group of units in Z ?₁₆
a) 4
b) 8
c) 12
d) 16
Answer: b) 8
91. In a field F, what is the minimal polynomial of √2 + √3 + √5?
a) x³ - 10x - 4
b) x - 20x⁴ + 80x² - 4⁶
c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
d) x² - 10
Answer: c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
92. Which of the following is not a property of Euclidean domains?
a) They are principal ideal domains
b) They have a division algorithm
c) They are always fields
d) They are unique factorization domains
Answer: c) They are always fields
93. What is the Galois group of x³ - 3x - 1 over Q?
a) Z₃
b) S₃
c) A₃
d) Trivial group
Answer: b) S₃
94. In a group G, what is the class equation?
a) |G| = |Z(G)| + Σ [G : C(x)]
b) |G| = |Z(G)| + Σ |C(x)|
c) |G| = Σ |orb(x)|
d) |G| = Π |Sylow p-subgroups|
Answer: a) |G| = |Z(G)| + Σ [G : C(x)]
95. Which of the following is an example of a non-cyclic simple group?
a) Z₅
b) A₄
c) Q₈
d) D₁₂
Answer: b) A₄
96. What is the transcendence degree of C over Q?
a) 0
b) 1
c) 2
d) Uncountably infinite
Answer: d) Uncountably infinite
97. In a ring R, what is the Jacobson radical?
a) The intersection of all maximal left ideals
b) The union of all minimal prime ideals
c) The set of all nilpotent elements
d) The largest nilpotent ideal
Answer: a) The intersection of all maximal left ideals
98. What is the fixed field of the Galois group Gal(Q(ζ )/Q), where ζ is a primitive 8th root of ₈ ₈
unity?
a) Q
b) Q(i)
c) Q(√2)
d) Q(√2, i)
Answer: c) Q(√2)
99. Which of the following is true for any finite simple group G?
a) |G| is always prime
b) G is always non-abelian except when |G| is prime
c) G is isomorphic to a subgroup of some symmetric group
d) G has no proper subgroups
Answer: b) G is always non-abelian except when |G| is prime
What is the multiplicative group of the field F ?₉
a) Z₈
b) Z₆
c) Z × Z₃ ₃
d) S₃
Answer: a) Z₈
62. In a ring R, what is a unit?
a) An element with a multiplicative inverse
b) An element with an additive inverse
c) An element that commutes with all other elements
d) An element that divides 1
Answer: a) An element with a multiplicative inverse
63. What is the Galois group of x⁴ - 1 over Q?
a) Z₂
b) Z₄
c) Z × Z₂ ₂
d) D₄
Answer: c) Z × Z₂ ₂
64. Which of the following is an example of a non-commutative division ring?
a) Q
b) R
c) C
d) Quaternions
Answer: d) Quaternions
65. What is the order of the group of automorphisms of Z ?₆
a) 1
b) 2
c) 3
d) 6
Answer: b) 2
66. In a field F, what is the minimal polynomial of a primitive 5th root of unity?
a) x - 1
b) x² + x + 1
c) x⁴ + x³ + x² + x + 1
d) x⁵ - 1
Answer: c) x⁴ + x³ + x² + x + 1
67. Which of the following is true for any finite field F?
a) |F| = p for some prime p
b) F is always an extension of Z₂
c) The multiplicative group F* is cyclic
d) F has characteristic 0
Answer: c) The multiplicative group F* is cyclic
68. What is the rank of the free abelian group Z × Z × Z?
a) 1
b) 2
c) 3
d) Infinite
Answer: c) 3
69. In a ring R, what is a prime ideal?
a) An ideal that contains only prime elements
b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
c) An ideal that is contained in every maximal ideal
d) An ideal generated by a single element
Answer: b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
70. What is the degree of the splitting field of x³ - 2 over Q?
a) 3
b) 4
c) 6
d) 8
Answer: c) 6
71. Which of the following is not a subring of Q[x]?
a) Z[x]
b) Q
c) Z
d) Q[x²]
Answer: b) Q
72. What is the center of the dihedral group D ?₈
a) {e}
b) {e, r²}
c) {e, r, r², r³}
d) D₈
Answer: b) {e, r²}
73. In a field F, what is the minimal polynomial of √2 + √3?
a) x² - 5
b) x² - 2x - 1
c) x⁴ - 10x² + 1
d) x⁴ - 5x² + 6
Answer: c) x⁴ - 10x² + 1
74. Which of the following is an example of a local ring?
a) Z
b) Q
c) Z/(p²) where p is prime
d) R[x]
Answer: c) Z/(p²) where p is prime
75. What is the order of GL (Z ), the group of invertible 3x3 matrices over Z ?₃ ₂ ₂
a) 24
b) 168
c) 336
d) 512
Answer: b) 168
76. In a group G, what is the normalizer of a subgroup H?
a) {g G | ghg ¹ H for all h H}∈ ⁻ ∈ ∈
b) {g G | gH = Hg}∈
c) {g G | gHg ¹ = H}∈ ⁻
d) All of the above
Answer: d) All of the above
77. What is the Krull dimension of the ring Z?
a) 0
b) 1
c) 2
d) Infinite
Answer: b) 1
78. Which of the following is true for any field extension E/F?
a) [E:F] = 1 if and only if E = F
b) If [E:F] is prime, then E/F is a simple extension
c) If [E:F] is finite, then E is algebraic over F
d) All of the above
Answer: d) All of the above
79. What is the order of the group Aut(Z ), the automorphism group of Z ?₈ ₈
a) 2
b) 4
c) 8
d) 16
Answer: b) 4
80. In a ring R, what is a maximal ideal?
a) An ideal that is not contained in any other proper ideal
b) An ideal that contains all other ideals
c) An ideal generated by a maximal element
d) An ideal with the largest number of elements
Answer: a) An ideal that is not contained in any other proper ideal
81. What is the Galois group of x⁴ + 1 over Q?
a) Z₄
b) Z × Z₂ ₂
c) D₄
d) S₄
Answer: c) D₄
82. Which of the following is an example of a non-Noetherian ring?
a) Z
b) Q[x]
c) R[[x]] (formal power series over R)
d) M (R) (2x2 matrices over R)₂
Answer: c) R[[x]] (formal power series over R)
83. What is the degree of the minimal polynomial of a primitive 7th root of unity over Q?
a) 3
b) 6
c) 7
d) 14
Answer: b) 6
84. In a group G, what is the commutator subgroup?
a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
b) The largest normal subgroup of G
c) The subgroup of all elements that commute with every element in G
d) The subgroup of all elements of order 2
Answer: a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
85. Which of the following is true for any finite-dimensional vector space V over a field F?
a) V is isomorphic to F^n for some n
b) V has a unique basis
c) All subspaces of V are finite-dimensional
d) The dual space V* is isomorphic to V
Answer: a) V is isomorphic to F^n for some n
86. What is the characteristic of the field Q(i)?
a) 0
b) 2
c) 4
d) Infinite
Answer: a) 0
87. In a ring R, what is a nilpotent element?
a) An element a such that a² = 0
b) An element a such that aⁿ = 0 for some positive integer n
c) An element a such that a² = a
d) An element a such that ab = 0 for some non-zero b in R
Answer: b) An element a such that aⁿ = 0 for some positive integer n
88. What is the Sylow 3-subgroup of A ?₄
a) Z₃
b) Z × Z₃ ₃
c) A₃
d) S₃
Answer: a) Z₃
89. Which of the following is an example of a field that is not algebraically closed?
a) C
b) R
c) Q
[ (algebraic closure of Q)
d) F
[ (algebraic closure of F )₂ ₂
Answer: b) R
90. What is the order of the group of units in Z ?₁₆
a) 4
b) 8
c) 12
d) 16
Answer: b) 8
91. In a field F, what is the minimal polynomial of √2 + √3 + √5?
a) x³ - 10x - 4
b) x - 20x⁴ + 80x² - 4⁶
c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
d) x² - 10
Answer: c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
92. Which of the following is not a property of Euclidean domains?
a) They are principal ideal domains
b) They have a division algorithm
c) They are always fields
d) They are unique factorization domains
Answer: c) They are always fields
93. What is the Galois group of x³ - 3x - 1 over Q?
a) Z₃
b) S₃
c) A₃
d) Trivial group
Answer: b) S₃
94. In a group G, what is the class equation?
a) |G| = |Z(G)| + Σ [G : C(x)]
b) |G| = |Z(G)| + Σ |C(x)|
c) |G| = Σ |orb(x)|
d) |G| = Π |Sylow p-subgroups|
Answer: a) |G| = |Z(G)| + Σ [G : C(x)]
95. Which of the following is an example of a non-cyclic simple group?
a) Z₅
b) A₄
c) Q₈
d) D₁₂
Answer: b) A₄
96. What is the transcendence degree of C over Q?
a) 0
b) 1
c) 2
d) Uncountably infinite
Answer: d) Uncountably infinite
97. In a ring R, what is the Jacobson radical?
a) The intersection of all maximal left ideals
b) The union of all minimal prime ideals
c) The set of all nilpotent elements
d) The largest nilpotent ideal
Answer: a) The intersection of all maximal left ideals
98. What is the fixed field of the Galois group Gal(Q(ζ )/Q), where ζ is a primitive 8th root of ₈ ₈
unity?
a) Q
b) Q(i)
c) Q(√2)
d) Q(√2, i)
Answer: c) Q(√2)
99. Which of the following is true for any finite simple group G?
a) |G| is always prime
b) G is always non-abelian except when |G| is prime
c) G is isomorphic to a subgroup of some symmetric group
d) G has no proper subgroups
Answer: b) G is always non-abelian except when |G| is prime
What is the multiplicative group of the field F ?₉
a) Z₈
b) Z₆
c) Z × Z₃ ₃
d) S₃
Answer: a) Z₈
62. In a ring R, what is a unit?
a) An element with a multiplicative inverse
b) An element with an additive inverse
c) An element that commutes with all other elements
d) An element that divides 1
Answer: a) An element with a multiplicative inverse
63. What is the Galois group of x⁴ - 1 over Q?
a) Z₂
b) Z₄
c) Z × Z₂ ₂
d) D₄
Answer: c) Z × Z₂ ₂
64. Which of the following is an example of a non-commutative division ring?
a) Q
b) R
c) C
d) Quaternions
Answer: d) Quaternions
65. What is the order of the group of automorphisms of Z ?₆
a) 1
b) 2
c) 3
d) 6
Answer: b) 2
66. In a field F, what is the minimal polynomial of a primitive 5th root of unity?
a) x - 1
b) x² + x + 1
c) x⁴ + x³ + x² + x + 1
d) x⁵ - 1
Answer: c) x⁴ + x³ + x² + x + 1
67. Which of the following is true for any finite field F?
a) |F| = p for some prime p
b) F is always an extension of Z₂
c) The multiplicative group F* is cyclic
d) F has characteristic 0
Answer: c) The multiplicative group F* is cyclic
68. What is the rank of the free abelian group Z × Z × Z?
a) 1
b) 2
c) 3
d) Infinite
Answer: c) 3
69. In a ring R, what is a prime ideal?
a) An ideal that contains only prime elements
b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
c) An ideal that is contained in every maximal ideal
d) An ideal generated by a single element
Answer: b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
70. What is the degree of the splitting field of x³ - 2 over Q?
a) 3
b) 4
c) 6
d) 8
Answer: c) 6
71. Which of the following is not a subring of Q[x]?
a) Z[x]
b) Q
c) Z
d) Q[x²]
Answer: b) Q
72. What is the center of the dihedral group D ?₈
a) {e}
b) {e, r²}
c) {e, r, r², r³}
d) D₈
Answer: b) {e, r²}
73. In a field F, what is the minimal polynomial of √2 + √3?
a) x² - 5
b) x² - 2x - 1
c) x⁴ - 10x² + 1
d) x⁴ - 5x² + 6
Answer: c) x⁴ - 10x² + 1
74. Which of the following is an example of a local ring?
a) Z
b) Q
c) Z/(p²) where p is prime
d) R[x]
Answer: c) Z/(p²) where p is prime
75. What is the order of GL (Z ), the group of invertible 3x3 matrices over Z ?₃ ₂ ₂
a) 24
b) 168
c) 336
d) 512
Answer: b) 168
76. In a group G, what is the normalizer of a subgroup H?
a) {g G | ghg ¹ H for all h H}∈ ⁻ ∈ ∈
b) {g G | gH = Hg}∈
c) {g G | gHg ¹ = H}∈ ⁻
d) All of the above
Answer: d) All of the above
77. What is the Krull dimension of the ring Z?
a) 0
b) 1
c) 2
d) Infinite
Answer: b) 1
78. Which of the following is true for any field extension E/F?
a) [E:F] = 1 if and only if E = F
b) If [E:F] is prime, then E/F is a simple extension
c) If [E:F] is finite, then E is algebraic over F
d) All of the above
Answer: d) All of the above
79. What is the order of the group Aut(Z ), the automorphism group of Z ?₈ ₈
a) 2
b) 4
c) 8
d) 16
Answer: b) 4
80. In a ring R, what is a maximal ideal?
a) An ideal that is not contained in any other proper ideal
b) An ideal that contains all other ideals
c) An ideal generated by a maximal element
d) An ideal with the largest number of elements
Answer: a) An ideal that is not contained in any other proper ideal
81. What is the Galois group of x⁴ + 1 over Q?
a) Z₄
b) Z × Z₂ ₂
c) D₄
d) S₄
Answer: c) D₄
82. Which of the following is an example of a non-Noetherian ring?
a) Z
b) Q[x]
c) R[[x]] (formal power series over R)
d) M (R) (2x2 matrices over R)₂
Answer: c) R[[x]] (formal power series over R)
83. What is the degree of the minimal polynomial of a primitive 7th root of unity over Q?
a) 3
b) 6
c) 7
d) 14
Answer: b) 6
84. In a group G, what is the commutator subgroup?
a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
b) The largest normal subgroup of G
c) The subgroup of all elements that commute with every element in G
d) The subgroup of all elements of order 2
Answer: a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
85. Which of the following is true for any finite-dimensional vector space V over a field F?
a) V is isomorphic to F^n for some n
b) V has a unique basis
c) All subspaces of V are finite-dimensional
d) The dual space V* is isomorphic to V
Answer: a) V is isomorphic to F^n for some n
86. What is the characteristic of the field Q(i)?
a) 0
b) 2
c) 4
d) Infinite
Answer: a) 0
87. In a ring R, what is a nilpotent element?
a) An element a such that a² = 0
b) An element a such that aⁿ = 0 for some positive integer n
c) An element a such that a² = a
d) An element a such that ab = 0 for some non-zero b in R
Answer: b) An element a such that aⁿ = 0 for some positive integer n
88. What is the Sylow 3-subgroup of A ?₄
a) Z₃
b) Z × Z₃ ₃
c) A₃
d) S₃
Answer: a) Z₃
89. Which of the following is an example of a field that is not algebraically closed?
a) C
b) R
c) Q
[ (algebraic closure of Q)
d) F
[ (algebraic closure of F )₂ ₂
Answer: b) R
90. What is the order of the group of units in Z ?₁₆
a) 4
b) 8
c) 12
d) 16
Answer: b) 8
91. In a field F, what is the minimal polynomial of √2 + √3 + √5?
a) x³ - 10x - 4
b) x - 20x⁴ + 80x² - 4⁶
c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
d) x² - 10
Answer: c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
92. Which of the following is not a property of Euclidean domains?
a) They are principal ideal domains
b) They have a division algorithm
c) They are always fields
d) They are unique factorization domains
Answer: c) They are always fields
93. What is the Galois group of x³ - 3x - 1 over Q?
a) Z₃
b) S₃
c) A₃
d) Trivial group
Answer: b) S₃
94. In a group G, what is the class equation?
a) |G| = |Z(G)| + Σ [G : C(x)]
b) |G| = |Z(G)| + Σ |C(x)|
c) |G| = Σ |orb(x)|
d) |G| = Π |Sylow p-subgroups|
Answer: a) |G| = |Z(G)| + Σ [G : C(x)]
95. Which of the following is an example of a non-cyclic simple group?
a) Z₅
b) A₄
c) Q₈
d) D₁₂
Answer: b) A₄
96. What is the transcendence degree of C over Q?
a) 0
b) 1
c) 2
d) Uncountably infinite
Answer: d) Uncountably infinite
97. In a ring R, what is the Jacobson radical?
a) The intersection of all maximal left ideals
b) The union of all minimal prime ideals
c) The set of all nilpotent elements
d) The largest nilpotent ideal
Answer: a) The intersection of all maximal left ideals
98. What is the fixed field of the Galois group Gal(Q(ζ )/Q), where ζ is a primitive 8th root of ₈ ₈
unity?
a) Q
b) Q(i)
c) Q(√2)
d) Q(√2, i)
Answer: c) Q(√2)
99. Which of the following is true for any finite simple group G?
a) |G| is always prime
b) G is always non-abelian except when |G| is prime
c) G is isomorphic to a subgroup of some symmetric group
d) G has no proper subgroups
Answer: b) G is always non-abelian except when |G| is prime
What is the multiplicative group of the field F ?₉
a) Z₈
b) Z₆
c) Z × Z₃ ₃
d) S₃
Answer: a) Z₈
62. In a ring R, what is a unit?
a) An element with a multiplicative inverse
b) An element with an additive inverse
c) An element that commutes with all other elements
d) An element that divides 1
Answer: a) An element with a multiplicative inverse
63. What is the Galois group of x⁴ - 1 over Q?
a) Z₂
b) Z₄
c) Z × Z₂ ₂
d) D₄
Answer: c) Z × Z₂ ₂
64. Which of the following is an example of a non-commutative division ring?
a) Q
b) R
c) C
d) Quaternions
Answer: d) Quaternions
65. What is the order of the group of automorphisms of Z ?₆
a) 1
b) 2
c) 3
d) 6
Answer: b) 2
66. In a field F, what is the minimal polynomial of a primitive 5th root of unity?
a) x - 1
b) x² + x + 1
c) x⁴ + x³ + x² + x + 1
d) x⁵ - 1
Answer: c) x⁴ + x³ + x² + x + 1
67. Which of the following is true for any finite field F?
a) |F| = p for some prime p
b) F is always an extension of Z₂
c) The multiplicative group F* is cyclic
d) F has characteristic 0
Answer: c) The multiplicative group F* is cyclic
68. What is the rank of the free abelian group Z × Z × Z?
a) 1
b) 2
c) 3
d) Infinite
Answer: c) 3
69. In a ring R, what is a prime ideal?
a) An ideal that contains only prime elements
b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
c) An ideal that is contained in every maximal ideal
d) An ideal generated by a single element
Answer: b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
70. What is the degree of the splitting field of x³ - 2 over Q?
a) 3
b) 4
c) 6
d) 8
Answer: c) 6
71. Which of the following is not a subring of Q[x]?
a) Z[x]
b) Q
c) Z
d) Q[x²]
Answer: b) Q
72. What is the center of the dihedral group D ?₈
a) {e}
b) {e, r²}
c) {e, r, r², r³}
d) D₈
Answer: b) {e, r²}
73. In a field F, what is the minimal polynomial of √2 + √3?
a) x² - 5
b) x² - 2x - 1
c) x⁴ - 10x² + 1
d) x⁴ - 5x² + 6
Answer: c) x⁴ - 10x² + 1
74. Which of the following is an example of a local ring?
a) Z
b) Q
c) Z/(p²) where p is prime
d) R[x]
Answer: c) Z/(p²) where p is prime
75. What is the order of GL (Z ), the group of invertible 3x3 matrices over Z ?₃ ₂ ₂
a) 24
b) 168
c) 336
d) 512
Answer: b) 168
76. In a group G, what is the normalizer of a subgroup H?
a) {g G | ghg ¹ H for all h H}∈ ⁻ ∈ ∈
b) {g G | gH = Hg}∈
c) {g G | gHg ¹ = H}∈ ⁻
d) All of the above
Answer: d) All of the above
77. What is the Krull dimension of the ring Z?
a) 0
b) 1
c) 2
d) Infinite
Answer: b) 1
78. Which of the following is true for any field extension E/F?
a) [E:F] = 1 if and only if E = F
b) If [E:F] is prime, then E/F is a simple extension
c) If [E:F] is finite, then E is algebraic over F
d) All of the above
Answer: d) All of the above
79. What is the order of the group Aut(Z ), the automorphism group of Z ?₈ ₈
a) 2
b) 4
c) 8
d) 16
Answer: b) 4
80. In a ring R, what is a maximal ideal?
a) An ideal that is not contained in any other proper ideal
b) An ideal that contains all other ideals
c) An ideal generated by a maximal element
d) An ideal with the largest number of elements
Answer: a) An ideal that is not contained in any other proper ideal
81. What is the Galois group of x⁴ + 1 over Q?
a) Z₄
b) Z × Z₂ ₂
c) D₄
d) S₄
Answer: c) D₄
82. Which of the following is an example of a non-Noetherian ring?
a) Z
b) Q[x]
c) R[[x]] (formal power series over R)
d) M (R) (2x2 matrices over R)₂
Answer: c) R[[x]] (formal power series over R)
83. What is the degree of the minimal polynomial of a primitive 7th root of unity over Q?
a) 3
b) 6
c) 7
d) 14
Answer: b) 6
84. In a group G, what is the commutator subgroup?
a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
b) The largest normal subgroup of G
c) The subgroup of all elements that commute with every element in G
d) The subgroup of all elements of order 2
Answer: a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
85. Which of the following is true for any finite-dimensional vector space V over a field F?
a) V is isomorphic to F^n for some n
b) V has a unique basis
c) All subspaces of V are finite-dimensional
d) The dual space V* is isomorphic to V
Answer: a) V is isomorphic to F^n for some n
86. What is the characteristic of the field Q(i)?
a) 0
b) 2
c) 4
d) Infinite
Answer: a) 0
87. In a ring R, what is a nilpotent element?
a) An element a such that a² = 0
b) An element a such that aⁿ = 0 for some positive integer n
c) An element a such that a² = a
d) An element a such that ab = 0 for some non-zero b in R
Answer: b) An element a such that aⁿ = 0 for some positive integer n
88. What is the Sylow 3-subgroup of A ?₄
a) Z₃
b) Z × Z₃ ₃
c) A₃
d) S₃
Answer: a) Z₃
89. Which of the following is an example of a field that is not algebraically closed?
a) C
b) R
c) Q
[ (algebraic closure of Q)
d) F
[ (algebraic closure of F )₂ ₂
Answer: b) R
90. What is the order of the group of units in Z ?₁₆
a) 4
b) 8
c) 12
d) 16
Answer: b) 8
91. In a field F, what is the minimal polynomial of √2 + √3 + √5?
a) x³ - 10x - 4
b) x - 20x⁴ + 80x² - 4⁶
c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
d) x² - 10
Answer: c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
92. Which of the following is not a property of Euclidean domains?
a) They are principal ideal domains
b) They have a division algorithm
c) They are always fields
d) They are unique factorization domains
Answer: c) They are always fields
93. What is the Galois group of x³ - 3x - 1 over Q?
a) Z₃
b) S₃
c) A₃
d) Trivial group
Answer: b) S₃
94. In a group G, what is the class equation?
a) |G| = |Z(G)| + Σ [G : C(x)]
b) |G| = |Z(G)| + Σ |C(x)|
c) |G| = Σ |orb(x)|
d) |G| = Π |Sylow p-subgroups|
Answer: a) |G| = |Z(G)| + Σ [G : C(x)]
95. Which of the following is an example of a non-cyclic simple group?
a) Z₅
b) A₄
c) Q₈
d) D₁₂
Answer: b) A₄
96. What is the transcendence degree of C over Q?
a) 0
b) 1
c) 2
d) Uncountably infinite
Answer: d) Uncountably infinite
97. In a ring R, what is the Jacobson radical?
a) The intersection of all maximal left ideals
b) The union of all minimal prime ideals
c) The set of all nilpotent elements
d) The largest nilpotent ideal
Answer: a) The intersection of all maximal left ideals
98. What is the fixed field of the Galois group Gal(Q(ζ )/Q), where ζ is a primitive 8th root of ₈ ₈
unity?
a) Q
b) Q(i)
c) Q(√2)
d) Q(√2, i)
Answer: c) Q(√2)
99. Which of the following is true for any finite simple group G?
a) |G| is always prime
b) G is always non-abelian except when |G| is prime
c) G is isomorphic to a subgroup of some symmetric group
d) G has no proper subgroups
Answer: b) G is always non-abelian except when |G| is prime
What is the multiplicative group of the field F ?₉
a) Z₈
b) Z₆
c) Z × Z₃ ₃
d) S₃
Answer: a) Z₈
62. In a ring R, what is a unit?
a) An element with a multiplicative inverse
b) An element with an additive inverse
c) An element that commutes with all other elements
d) An element that divides 1
Answer: a) An element with a multiplicative inverse
63. What is the Galois group of x⁴ - 1 over Q?
a) Z₂
b) Z₄
c) Z × Z₂ ₂
d) D₄
Answer: c) Z × Z₂ ₂
64. Which of the following is an example of a non-commutative division ring?
a) Q
b) R
c) C
d) Quaternions
Answer: d) Quaternions
65. What is the order of the group of automorphisms of Z ?₆
a) 1
b) 2
c) 3
d) 6
Answer: b) 2
66. In a field F, what is the minimal polynomial of a primitive 5th root of unity?
a) x - 1
b) x² + x + 1
c) x⁴ + x³ + x² + x + 1
d) x⁵ - 1
Answer: c) x⁴ + x³ + x² + x + 1
67. Which of the following is true for any finite field F?
a) |F| = p for some prime p
b) F is always an extension of Z₂
c) The multiplicative group F* is cyclic
d) F has characteristic 0
Answer: c) The multiplicative group F* is cyclic
68. What is the rank of the free abelian group Z × Z × Z?
a) 1
b) 2
c) 3
d) Infinite
Answer: c) 3
69. In a ring R, what is a prime ideal?
a) An ideal that contains only prime elements
b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
c) An ideal that is contained in every maximal ideal
d) An ideal generated by a single element
Answer: b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
70. What is the degree of the splitting field of x³ - 2 over Q?
a) 3
b) 4
c) 6
d) 8
Answer: c) 6
71. Which of the following is not a subring of Q[x]?
a) Z[x]
b) Q
c) Z
d) Q[x²]
Answer: b) Q
72. What is the center of the dihedral group D ?₈
a) {e}
b) {e, r²}
c) {e, r, r², r³}
d) D₈
Answer: b) {e, r²}
73. In a field F, what is the minimal polynomial of √2 + √3?
a) x² - 5
b) x² - 2x - 1
c) x⁴ - 10x² + 1
d) x⁴ - 5x² + 6
Answer: c) x⁴ - 10x² + 1
74. Which of the following is an example of a local ring?
a) Z
b) Q
c) Z/(p²) where p is prime
d) R[x]
Answer: c) Z/(p²) where p is prime
75. What is the order of GL (Z ), the group of invertible 3x3 matrices over Z ?₃ ₂ ₂
a) 24
b) 168
c) 336
d) 512
Answer: b) 168
76. In a group G, what is the normalizer of a subgroup H?
a) {g G | ghg ¹ H for all h H}∈ ⁻ ∈ ∈
b) {g G | gH = Hg}∈
c) {g G | gHg ¹ = H}∈ ⁻
d) All of the above
Answer: d) All of the above
77. What is the Krull dimension of the ring Z?
a) 0
b) 1
c) 2
d) Infinite
Answer: b) 1
78. Which of the following is true for any field extension E/F?
a) [E:F] = 1 if and only if E = F
b) If [E:F] is prime, then E/F is a simple extension
c) If [E:F] is finite, then E is algebraic over F
d) All of the above
Answer: d) All of the above
79. What is the order of the group Aut(Z ), the automorphism group of Z ?₈ ₈
a) 2
b) 4
c) 8
d) 16
Answer: b) 4
80. In a ring R, what is a maximal ideal?
a) An ideal that is not contained in any other proper ideal
b) An ideal that contains all other ideals
c) An ideal generated by a maximal element
d) An ideal with the largest number of elements
Answer: a) An ideal that is not contained in any other proper ideal
81. What is the Galois group of x⁴ + 1 over Q?
a) Z₄
b) Z × Z₂ ₂
c) D₄
d) S₄
Answer: c) D₄
82. Which of the following is an example of a non-Noetherian ring?
a) Z
b) Q[x]
c) R[[x]] (formal power series over R)
d) M (R) (2x2 matrices over R)₂
Answer: c) R[[x]] (formal power series over R)
83. What is the degree of the minimal polynomial of a primitive 7th root of unity over Q?
a) 3
b) 6
c) 7
d) 14
Answer: b) 6
84. In a group G, what is the commutator subgroup?
a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
b) The largest normal subgroup of G
c) The subgroup of all elements that commute with every element in G
d) The subgroup of all elements of order 2
Answer: a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
85. Which of the following is true for any finite-dimensional vector space V over a field F?
a) V is isomorphic to F^n for some n
b) V has a unique basis
c) All subspaces of V are finite-dimensional
d) The dual space V* is isomorphic to V
Answer: a) V is isomorphic to F^n for some n
86. What is the characteristic of the field Q(i)?
a) 0
b) 2
c) 4
d) Infinite
Answer: a) 0
87. In a ring R, what is a nilpotent element?
a) An element a such that a² = 0
b) An element a such that aⁿ = 0 for some positive integer n
c) An element a such that a² = a
d) An element a such that ab = 0 for some non-zero b in R
Answer: b) An element a such that aⁿ = 0 for some positive integer n
88. What is the Sylow 3-subgroup of A ?₄
a) Z₃
b) Z × Z₃ ₃
c) A₃
d) S₃
Answer: a) Z₃
89. Which of the following is an example of a field that is not algebraically closed?
a) C
b) R
c) Q
[ (algebraic closure of Q)
d) F
[ (algebraic closure of F )₂ ₂
Answer: b) R
90. What is the order of the group of units in Z ?₁₆
a) 4
b) 8
c) 12
d) 16
Answer: b) 8
91. In a field F, what is the minimal polynomial of √2 + √3 + √5?
a) x³ - 10x - 4
b) x - 20x⁴ + 80x² - 4⁶
c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
d) x² - 10
Answer: c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
92. Which of the following is not a property of Euclidean domains?
a) They are principal ideal domains
b) They have a division algorithm
c) They are always fields
d) They are unique factorization domains
Answer: c) They are always fields
93. What is the Galois group of x³ - 3x - 1 over Q?
a) Z₃
b) S₃
c) A₃
d) Trivial group
Answer: b) S₃
94. In a group G, what is the class equation?
a) |G| = |Z(G)| + Σ [G : C(x)]
b) |G| = |Z(G)| + Σ |C(x)|
c) |G| = Σ |orb(x)|
d) |G| = Π |Sylow p-subgroups|
Answer: a) |G| = |Z(G)| + Σ [G : C(x)]
95. Which of the following is an example of a non-cyclic simple group?
a) Z₅
b) A₄
c) Q₈
d) D₁₂
Answer: b) A₄
96. What is the transcendence degree of C over Q?
a) 0
b) 1
c) 2
d) Uncountably infinite
Answer: d) Uncountably infinite
97. In a ring R, what is the Jacobson radical?
a) The intersection of all maximal left ideals
b) The union of all minimal prime ideals
c) The set of all nilpotent elements
d) The largest nilpotent ideal
Answer: a) The intersection of all maximal left ideals
98. What is the fixed field of the Galois group Gal(Q(ζ )/Q), where ζ is a primitive 8th root of ₈ ₈
unity?
a) Q
b) Q(i)
c) Q(√2)
d) Q(√2, i)
Answer: c) Q(√2)
99. Which of the following is true for any finite simple group G?
a) |G| is always prime
b) G is always non-abelian except when |G| is prime
c) G is isomorphic to a subgroup of some symmetric group
d) G has no proper subgroups
Answer: b) G is always non-abelian except when |G| is prime
What is the multiplicative group of the field F ?₉
a) Z₈
b) Z₆
c) Z × Z₃ ₃
d) S₃
Answer: a) Z₈
62. In a ring R, what is a unit?
a) An element with a multiplicative inverse
b) An element with an additive inverse
c) An element that commutes with all other elements
d) An element that divides 1
Answer: a) An element with a multiplicative inverse
63. What is the Galois group of x⁴ - 1 over Q?
a) Z₂
b) Z₄
c) Z × Z₂ ₂
d) D₄
Answer: c) Z × Z₂ ₂
64. Which of the following is an example of a non-commutative division ring?
a) Q
b) R
c) C
d) Quaternions
Answer: d) Quaternions
65. What is the order of the group of automorphisms of Z ?₆
a) 1
b) 2
c) 3
d) 6
Answer: b) 2
66. In a field F, what is the minimal polynomial of a primitive 5th root of unity?
a) x - 1
b) x² + x + 1
c) x⁴ + x³ + x² + x + 1
d) x⁵ - 1
Answer: c) x⁴ + x³ + x² + x + 1
67. Which of the following is true for any finite field F?
a) |F| = p for some prime p
b) F is always an extension of Z₂
c) The multiplicative group F* is cyclic
d) F has characteristic 0
Answer: c) The multiplicative group F* is cyclic
68. What is the rank of the free abelian group Z × Z × Z?
a) 1
b) 2
c) 3
d) Infinite
Answer: c) 3
69. In a ring R, what is a prime ideal?
a) An ideal that contains only prime elements
b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
c) An ideal that is contained in every maximal ideal
d) An ideal generated by a single element
Answer: b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
70. What is the degree of the splitting field of x³ - 2 over Q?
a) 3
b) 4
c) 6
d) 8
Answer: c) 6
71. Which of the following is not a subring of Q[x]?
a) Z[x]
b) Q
c) Z
d) Q[x²]
Answer: b) Q
72. What is the center of the dihedral group D ?₈
a) {e}
b) {e, r²}
c) {e, r, r², r³}
d) D₈
Answer: b) {e, r²}
73. In a field F, what is the minimal polynomial of √2 + √3?
a) x² - 5
b) x² - 2x - 1
c) x⁴ - 10x² + 1
d) x⁴ - 5x² + 6
Answer: c) x⁴ - 10x² + 1
74. Which of the following is an example of a local ring?
a) Z
b) Q
c) Z/(p²) where p is prime
d) R[x]
Answer: c) Z/(p²) where p is prime
75. What is the order of GL (Z ), the group of invertible 3x3 matrices over Z ?₃ ₂ ₂
a) 24
b) 168
c) 336
d) 512
Answer: b) 168
76. In a group G, what is the normalizer of a subgroup H?
a) {g G | ghg ¹ H for all h H}∈ ⁻ ∈ ∈
b) {g G | gH = Hg}∈
c) {g G | gHg ¹ = H}∈ ⁻
d) All of the above
Answer: d) All of the above
77. What is the Krull dimension of the ring Z?
a) 0
b) 1
c) 2
d) Infinite
Answer: b) 1
78. Which of the following is true for any field extension E/F?
a) [E:F] = 1 if and only if E = F
b) If [E:F] is prime, then E/F is a simple extension
c) If [E:F] is finite, then E is algebraic over F
d) All of the above
Answer: d) All of the above
79. What is the order of the group Aut(Z ), the automorphism group of Z ?₈ ₈
a) 2
b) 4
c) 8
d) 16
Answer: b) 4
80. In a ring R, what is a maximal ideal?
a) An ideal that is not contained in any other proper ideal
b) An ideal that contains all other ideals
c) An ideal generated by a maximal element
d) An ideal with the largest number of elements
Answer: a) An ideal that is not contained in any other proper ideal
81. What is the Galois group of x⁴ + 1 over Q?
a) Z₄
b) Z × Z₂ ₂
c) D₄
d) S₄
Answer: c) D₄
82. Which of the following is an example of a non-Noetherian ring?
a) Z
b) Q[x]
c) R[[x]] (formal power series over R)
d) M (R) (2x2 matrices over R)₂
Answer: c) R[[x]] (formal power series over R)
83. What is the degree of the minimal polynomial of a primitive 7th root of unity over Q?
a) 3
b) 6
c) 7
d) 14
Answer: b) 6
84. In a group G, what is the commutator subgroup?
a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
b) The largest normal subgroup of G
c) The subgroup of all elements that commute with every element in G
d) The subgroup of all elements of order 2
Answer: a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
85. Which of the following is true for any finite-dimensional vector space V over a field F?
a) V is isomorphic to F^n for some n
b) V has a unique basis
c) All subspaces of V are finite-dimensional
d) The dual space V* is isomorphic to V
Answer: a) V is isomorphic to F^n for some n
86. What is the characteristic of the field Q(i)?
a) 0
b) 2
c) 4
d) Infinite
Answer: a) 0
87. In a ring R, what is a nilpotent element?
a) An element a such that a² = 0
b) An element a such that aⁿ = 0 for some positive integer n
c) An element a such that a² = a
d) An element a such that ab = 0 for some non-zero b in R
Answer: b) An element a such that aⁿ = 0 for some positive integer n
88. What is the Sylow 3-subgroup of A ?₄
a) Z₃
b) Z × Z₃ ₃
c) A₃
d) S₃
Answer: a) Z₃
89. Which of the following is an example of a field that is not algebraically closed?
a) C
b) R
c) Q
[ (algebraic closure of Q)
d) F
[ (algebraic closure of F )₂ ₂
Answer: b) R
90. What is the order of the group of units in Z ?₁₆
a) 4
b) 8
c) 12
d) 16
Answer: b) 8
91. In a field F, what is the minimal polynomial of √2 + √3 + √5?
a) x³ - 10x - 4
b) x - 20x⁴ + 80x² - 4⁶
c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
d) x² - 10
Answer: c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
92. Which of the following is not a property of Euclidean domains?
a) They are principal ideal domains
b) They have a division algorithm
c) They are always fields
d) They are unique factorization domains
Answer: c) They are always fields
93. What is the Galois group of x³ - 3x - 1 over Q?
a) Z₃
b) S₃
c) A₃
d) Trivial group
Answer: b) S₃
94. In a group G, what is the class equation?
a) |G| = |Z(G)| + Σ [G : C(x)]
b) |G| = |Z(G)| + Σ |C(x)|
c) |G| = Σ |orb(x)|
d) |G| = Π |Sylow p-subgroups|
Answer: a) |G| = |Z(G)| + Σ [G : C(x)]
95. Which of the following is an example of a non-cyclic simple group?
a) Z₅
b) A₄
c) Q₈
d) D₁₂
Answer: b) A₄
96. What is the transcendence degree of C over Q?
a) 0
b) 1
c) 2
d) Uncountably infinite
Answer: d) Uncountably infinite
97. In a ring R, what is the Jacobson radical?
a) The intersection of all maximal left ideals
b) The union of all minimal prime ideals
c) The set of all nilpotent elements
d) The largest nilpotent ideal
Answer: a) The intersection of all maximal left ideals
98. What is the fixed field of the Galois group Gal(Q(ζ )/Q), where ζ is a primitive 8th root of ₈ ₈
unity?
a) Q
b) Q(i)
c) Q(√2)
d) Q(√2, i)
Answer: c) Q(√2)
99. Which of the following is true for any finite simple group G?
a) |G| is always prime
b) G is always non-abelian except when |G| is prime
c) G is isomorphic to a subgroup of some symmetric group
d) G has no proper subgroups
Answer: b) G is always non-abelian except when |G| is prime
What is the multiplicative group of the field F ?₉
a) Z₈
b) Z₆
c) Z × Z₃ ₃
d) S₃
Answer: a) Z₈
62. In a ring R, what is a unit?
a) An element with a multiplicative inverse
b) An element with an additive inverse
c) An element that commutes with all other elements
d) An element that divides 1
Answer: a) An element with a multiplicative inverse
63. What is the Galois group of x⁴ - 1 over Q?
a) Z₂
b) Z₄
c) Z × Z₂ ₂
d) D₄
Answer: c) Z × Z₂ ₂
64. Which of the following is an example of a non-commutative division ring?
a) Q
b) R
c) C
d) Quaternions
Answer: d) Quaternions
65. What is the order of the group of automorphisms of Z ?₆
a) 1
b) 2
c) 3
d) 6
Answer: b) 2
66. In a field F, what is the minimal polynomial of a primitive 5th root of unity?
a) x - 1
b) x² + x + 1
c) x⁴ + x³ + x² + x + 1
d) x⁵ - 1
Answer: c) x⁴ + x³ + x² + x + 1
67. Which of the following is true for any finite field F?
a) |F| = p for some prime p
b) F is always an extension of Z₂
c) The multiplicative group F* is cyclic
d) F has characteristic 0
Answer: c) The multiplicative group F* is cyclic
68. What is the rank of the free abelian group Z × Z × Z?
a) 1
b) 2
c) 3
d) Infinite
Answer: c) 3
69. In a ring R, what is a prime ideal?
a) An ideal that contains only prime elements
b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
c) An ideal that is contained in every maximal ideal
d) An ideal generated by a single element
Answer: b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
70. What is the degree of the splitting field of x³ - 2 over Q?
a) 3
b) 4
c) 6
d) 8
Answer: c) 6
71. Which of the following is not a subring of Q[x]?
a) Z[x]
b) Q
c) Z
d) Q[x²]
Answer: b) Q
72. What is the center of the dihedral group D ?₈
a) {e}
b) {e, r²}
c) {e, r, r², r³}
d) D₈
Answer: b) {e, r²}
73. In a field F, what is the minimal polynomial of √2 + √3?
a) x² - 5
b) x² - 2x - 1
c) x⁴ - 10x² + 1
d) x⁴ - 5x² + 6
Answer: c) x⁴ - 10x² + 1
74. Which of the following is an example of a local ring?
a) Z
b) Q
c) Z/(p²) where p is prime
d) R[x]
Answer: c) Z/(p²) where p is prime
75. What is the order of GL (Z ), the group of invertible 3x3 matrices over Z ?₃ ₂ ₂
a) 24
b) 168
c) 336
d) 512
Answer: b) 168
76. In a group G, what is the normalizer of a subgroup H?
a) {g G | ghg ¹ H for all h H}∈ ⁻ ∈ ∈
b) {g G | gH = Hg}∈
c) {g G | gHg ¹ = H}∈ ⁻
d) All of the above
Answer: d) All of the above
77. What is the Krull dimension of the ring Z?
a) 0
b) 1
c) 2
d) Infinite
Answer: b) 1
78. Which of the following is true for any field extension E/F?
a) [E:F] = 1 if and only if E = F
b) If [E:F] is prime, then E/F is a simple extension
c) If [E:F] is finite, then E is algebraic over F
d) All of the above
Answer: d) All of the above
79. What is the order of the group Aut(Z ), the automorphism group of Z ?₈ ₈
a) 2
b) 4
c) 8
d) 16
Answer: b) 4
80. In a ring R, what is a maximal ideal?
a) An ideal that is not contained in any other proper ideal
b) An ideal that contains all other ideals
c) An ideal generated by a maximal element
d) An ideal with the largest number of elements
Answer: a) An ideal that is not contained in any other proper ideal
81. What is the Galois group of x⁴ + 1 over Q?
a) Z₄
b) Z × Z₂ ₂
c) D₄
d) S₄
Answer: c) D₄
82. Which of the following is an example of a non-Noetherian ring?
a) Z
b) Q[x]
c) R[[x]] (formal power series over R)
d) M (R) (2x2 matrices over R)₂
Answer: c) R[[x]] (formal power series over R)
83. What is the degree of the minimal polynomial of a primitive 7th root of unity over Q?
a) 3
b) 6
c) 7
d) 14
Answer: b) 6
84. In a group G, what is the commutator subgroup?
a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
b) The largest normal subgroup of G
c) The subgroup of all elements that commute with every element in G
d) The subgroup of all elements of order 2
Answer: a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
85. Which of the following is true for any finite-dimensional vector space V over a field F?
a) V is isomorphic to F^n for some n
b) V has a unique basis
c) All subspaces of V are finite-dimensional
d) The dual space V* is isomorphic to V
Answer: a) V is isomorphic to F^n for some n
86. What is the characteristic of the field Q(i)?
a) 0
b) 2
c) 4
d) Infinite
Answer: a) 0
87. In a ring R, what is a nilpotent element?
a) An element a such that a² = 0
b) An element a such that aⁿ = 0 for some positive integer n
c) An element a such that a² = a
d) An element a such that ab = 0 for some non-zero b in R
Answer: b) An element a such that aⁿ = 0 for some positive integer n
88. What is the Sylow 3-subgroup of A ?₄
a) Z₃
b) Z × Z₃ ₃
c) A₃
d) S₃
Answer: a) Z₃
89. Which of the following is an example of a field that is not algebraically closed?
a) C
b) R
c) Q
[ (algebraic closure of Q)
d) F
[ (algebraic closure of F )₂ ₂
Answer: b) R
90. What is the order of the group of units in Z ?₁₆
a) 4
b) 8
c) 12
d) 16
Answer: b) 8
91. In a field F, what is the minimal polynomial of √2 + √3 + √5?
a) x³ - 10x - 4
b) x - 20x⁴ + 80x² - 4⁶
c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
d) x² - 10
Answer: c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
92. Which of the following is not a property of Euclidean domains?
a) They are principal ideal domains
b) They have a division algorithm
c) They are always fields
d) They are unique factorization domains
Answer: c) They are always fields
93. What is the Galois group of x³ - 3x - 1 over Q?
a) Z₃
b) S₃
c) A₃
d) Trivial group
Answer: b) S₃
94. In a group G, what is the class equation?
a) |G| = |Z(G)| + Σ [G : C(x)]
b) |G| = |Z(G)| + Σ |C(x)|
c) |G| = Σ |orb(x)|
d) |G| = Π |Sylow p-subgroups|
Answer: a) |G| = |Z(G)| + Σ [G : C(x)]
95. Which of the following is an example of a non-cyclic simple group?
a) Z₅
b) A₄
c) Q₈
d) D₁₂
Answer: b) A₄
96. What is the transcendence degree of C over Q?
a) 0
b) 1
c) 2
d) Uncountably infinite
Answer: d) Uncountably infinite
97. In a ring R, what is the Jacobson radical?
a) The intersection of all maximal left ideals
b) The union of all minimal prime ideals
c) The set of all nilpotent elements
d) The largest nilpotent ideal
Answer: a) The intersection of all maximal left ideals
98. What is the fixed field of the Galois group Gal(Q(ζ )/Q), where ζ is a primitive 8th root of ₈ ₈
unity?
a) Q
b) Q(i)
c) Q(√2)
d) Q(√2, i)
Answer: c) Q(√2)
99. Which of the following is true for any finite simple group G?
a) |G| is always prime
b) G is always non-abelian except when |G| is prime
c) G is isomorphic to a subgroup of some symmetric group
d) G has no proper subgroups
Answer: b) G is always non-abelian except when |G| is prime
What is the multiplicative group of the field F ?₉
a) Z₈
b) Z₆
c) Z × Z₃ ₃
d) S₃
Answer: a) Z₈
62. In a ring R, what is a unit?
a) An element with a multiplicative inverse
b) An element with an additive inverse
c) An element that commutes with all other elements
d) An element that divides 1
Answer: a) An element with a multiplicative inverse
63. What is the Galois group of x⁴ - 1 over Q?
a) Z₂
b) Z₄
c) Z × Z₂ ₂
d) D₄
Answer: c) Z × Z₂ ₂
64. Which of the following is an example of a non-commutative division ring?
a) Q
b) R
c) C
d) Quaternions
Answer: d) Quaternions
65. What is the order of the group of automorphisms of Z ?₆
a) 1
b) 2
c) 3
d) 6
Answer: b) 2
66. In a field F, what is the minimal polynomial of a primitive 5th root of unity?
a) x - 1
b) x² + x + 1
c) x⁴ + x³ + x² + x + 1
d) x⁵ - 1
Answer: c) x⁴ + x³ + x² + x + 1
67. Which of the following is true for any finite field F?
a) |F| = p for some prime p
b) F is always an extension of Z₂
c) The multiplicative group F* is cyclic
d) F has characteristic 0
Answer: c) The multiplicative group F* is cyclic
68. What is the rank of the free abelian group Z × Z × Z?
a) 1
b) 2
c) 3
d) Infinite
Answer: c) 3
69. In a ring R, what is a prime ideal?
a) An ideal that contains only prime elements
b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
c) An ideal that is contained in every maximal ideal
d) An ideal generated by a single element
Answer: b) An ideal P such that if ab P, then a P or b P∈ ∈ ∈
70. What is the degree of the splitting field of x³ - 2 over Q?
a) 3
b) 4
c) 6
d) 8
Answer: c) 6
71. Which of the following is not a subring of Q[x]?
a) Z[x]
b) Q
c) Z
d) Q[x²]
Answer: b) Q
72. What is the center of the dihedral group D ?₈
a) {e}
b) {e, r²}
c) {e, r, r², r³}
d) D₈
Answer: b) {e, r²}
73. In a field F, what is the minimal polynomial of √2 + √3?
a) x² - 5
b) x² - 2x - 1
c) x⁴ - 10x² + 1
d) x⁴ - 5x² + 6
Answer: c) x⁴ - 10x² + 1
74. Which of the following is an example of a local ring?
a) Z
b) Q
c) Z/(p²) where p is prime
d) R[x]
Answer: c) Z/(p²) where p is prime
75. What is the order of GL (Z ), the group of invertible 3x3 matrices over Z ?₃ ₂ ₂
a) 24
b) 168
c) 336
d) 512
Answer: b) 168
76. In a group G, what is the normalizer of a subgroup H?
a) {g G | ghg ¹ H for all h H}∈ ⁻ ∈ ∈
b) {g G | gH = Hg}∈
c) {g G | gHg ¹ = H}∈ ⁻
d) All of the above
Answer: d) All of the above
77. What is the Krull dimension of the ring Z?
a) 0
b) 1
c) 2
d) Infinite
Answer: b) 1
78. Which of the following is true for any field extension E/F?
a) [E:F] = 1 if and only if E = F
b) If [E:F] is prime, then E/F is a simple extension
c) If [E:F] is finite, then E is algebraic over F
d) All of the above
Answer: d) All of the above
79. What is the order of the group Aut(Z ), the automorphism group of Z ?₈ ₈
a) 2
b) 4
c) 8
d) 16
Answer: b) 4
80. In a ring R, what is a maximal ideal?
a) An ideal that is not contained in any other proper ideal
b) An ideal that contains all other ideals
c) An ideal generated by a maximal element
d) An ideal with the largest number of elements
Answer: a) An ideal that is not contained in any other proper ideal
81. What is the Galois group of x⁴ + 1 over Q?
a) Z₄
b) Z × Z₂ ₂
c) D₄
d) S₄
Answer: c) D₄
82. Which of the following is an example of a non-Noetherian ring?
a) Z
b) Q[x]
c) R[[x]] (formal power series over R)
d) M (R) (2x2 matrices over R)₂
Answer: c) R[[x]] (formal power series over R)
83. What is the degree of the minimal polynomial of a primitive 7th root of unity over Q?
a) 3
b) 6
c) 7
d) 14
Answer: b) 6
84. In a group G, what is the commutator subgroup?
a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
b) The largest normal subgroup of G
c) The subgroup of all elements that commute with every element in G
d) The subgroup of all elements of order 2
Answer: a) The subgroup generated by all elements of the form aba ¹b ¹⁻ ⁻
85. Which of the following is true for any finite-dimensional vector space V over a field F?
a) V is isomorphic to F^n for some n
b) V has a unique basis
c) All subspaces of V are finite-dimensional
d) The dual space V* is isomorphic to V
Answer: a) V is isomorphic to F^n for some n
86. What is the characteristic of the field Q(i)?
a) 0
b) 2
c) 4
d) Infinite
Answer: a) 0
87. In a ring R, what is a nilpotent element?
a) An element a such that a² = 0
b) An element a such that aⁿ = 0 for some positive integer n
c) An element a such that a² = a
d) An element a such that ab = 0 for some non-zero b in R
Answer: b) An element a such that aⁿ = 0 for some positive integer n
88. What is the Sylow 3-subgroup of A ?₄
a) Z₃
b) Z × Z₃ ₃
c) A₃
d) S₃
Answer: a) Z₃
89. Which of the following is an example of a field that is not algebraically closed?
a) C
b) R
c) Q
[ (algebraic closure of Q)
d) F
[ (algebraic closure of F )₂ ₂
Answer: b) R
90. What is the order of the group of units in Z ?₁₆
a) 4
b) 8
c) 12
d) 16
Answer: b) 8
91. In a field F, what is the minimal polynomial of √2 + √3 + √5?
a) x³ - 10x - 4
b) x - 20x⁴ + 80x² - 4⁶
c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
d) x² - 10
Answer: c) x⁸ - 40x + 352x⁴ - 960x² + 576⁶
92. Which of the following is not a property of Euclidean domains?
a) They are principal ideal domains
b) They have a division algorithm
c) They are always fields
d) They are unique factorization domains
Answer: c) They are always fields
93. What is the Galois group of x³ - 3x - 1 over Q?
a) Z₃
b) S₃
c) A₃
d) Trivial group
Answer: b) S₃
94. In a group G, what is the class equation?
a) |G| = |Z(G)| + Σ [G : C(x)]
b) |G| = |Z(G)| + Σ |C(x)|
c) |G| = Σ |orb(x)|
d) |G| = Π |Sylow p-subgroups|
Answer: a) |G| = |Z(G)| + Σ [G : C(x)]
95. Which of the following is an example of a non-cyclic simple group?
a) Z₅
b) A₄
c) Q₈
d) D₁₂
Answer: b) A₄
96. What is the transcendence degree of C over Q?
a) 0
b) 1
c) 2
d) Uncountably infinite
Answer: d) Uncountably infinite
97. In a ring R, what is the Jacobson radical?
a) The intersection of all maximal left ideals
b) The union of all minimal prime ideals
c) The set of all nilpotent elements
d) The largest nilpotent ideal
Answer: a) The intersection of all maximal left ideals
98. What is the fixed field of the Galois group Gal(Q(ζ )/Q), where ζ is a primitive 8th root of ₈ ₈
unity?
a) Q
b) Q(i)
c) Q(√2)
d) Q(√2, i)
Answer: c) Q(√2)
99. Which of the following is true for any finite simple group G?
a) |G| is always prime
b) G is always non-abelian except when |G| is prime
c) G is isomorphic to a subgroup of some symmetric group
d) G has no proper subgroups
Answer: b) G is always non-abelian except when |G| is prime
100. What is the genus of the field Q(√2, √3, √5)?
a) 1
b) 2
c) 3
d) 4
Answer: c) 3
