Abstract Algebra: Groups, Rings, Fields, and Basic Algebraic Structures:
Multiple Choice Questions with Solutions
1. What is the order of the cyclic group generated by 7 in Z/20Z?
a) 7 b) 10 c) 14 d) 20
Answer: b) 10
Solution: Find the smallest positive integer n such that 7n ≡ 0 (mod 20). This occurs when n = 10.
2. In the symmetric group S4, what is the order of (1 2 3)?
a) 2 b) 3 c) 4 d) 6
Answer: b) 3
Solution: The order of a cycle is equal to its length. (1 2 3) is a 3-cycle.
3. Which of the following is not a subgroup of (R, +)?
a) Z b) Q c) {0} d) R*
Answer: d) R*
Solution: R* (non-zero real numbers) is not closed under addition, so it's not a subgroup of (R, +).
4. What is the center of the quaternion group Q8?
a) {1, -1} b) {1, -1, i, -i} c) {1, -1, j, -j} d) {1, -1, k, -k}
Answer: a) {1, -1}
Solution: The center consists of elements that commute with all others. Only 1 and -1 have this
property in Q8.
5. In Z/6Z, what is 4^(-1)?
a) 2 b) 3 c) 4 d) 5
Answer: c) 4
Solution: We need to find x such that 4x ≡ 1 (mod 6). This is satisfied by x = 4.
6. What is the order of GL(2, R), the general linear group of 2x2 matrices over R?
a) Finite b) Countably infinite c) Uncountably infinite d) Zero
Answer: c) Uncountably infinite
Solution: There are uncountably many 2x2 invertible matrices over R.
7. Which of the following is not an ideal in Z?
a) 2Z b) 3Z c) Z d) Z+
Answer: d) Z+
Solution: Z+ (positive integers) is not closed under subtraction, so it's not an ideal.
8. What is the characteristic of the field Z/5Z?
a) 0 b) 2 c) 5 d) Infinite
Answer: c) 5
Solution: The characteristic is the smallest positive integer n such that n·1 = 0 in the field.
9. How many elements are in the quotient group Z/12Z?
a) 6 b) 12 c) 24 d) Infinite
Answer: b) 12
Solution: The order of a quotient group is equal to the order of the original group divided by the
order of the subgroup.
10. What is the degree of the minimal polynomial of √2 over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 - 2, which has degree 2.
11. Which of the following is not a ring?
a) Z b) Q c) R d) C*
Answer: d) C*
Solution: C* (non-zero complex numbers) is not closed under addition, so it's not a ring.
12. What is the order of the alternating group A4?
a) 8 b) 12 c) 24 d) 48
Answer: b) 12
Solution: The order of An is n!/2. For n = 4, this is 4!/2 = 12.
13. In the ring Z[i], what is the norm of 3 + 4i?
a) 5 b) 7 c) 25 d) 49
Answer: c) 25
Solution: The norm of a + bi is a^2 + b^2. Here, 3^2 + 4^2 = 25.
14. What is the number of Sylow 2-subgroups in a group of order 24?
a) 1 b) 3 c) 4 d) 8
Answer: b) 3
Solution: By Sylow's theorems, the number of Sylow 2-subgroups must divide 3 and be congruent
to 1 mod 2.
15. Which of the following is not a field?
a) Q b) R c) C d) Z
Answer: d) Z
Solution: Z (integers) does not have multiplicative inverses for all non-zero elements, so it's not a
field.
16. What is the order of the automorphism group of Z/8Z?
a) 2 b) 4 c) 8 d) 16
Answer: b) 4
Solution: The automorphism group is isomorphic to (Z/8Z)*, which has order 4.
17. In the ring Z[x], what is the degree of (x^2 + 1)(x^3 - 2x + 1)?
a) 3 b) 4 c) 5 d) 6
Answer: c) 5
Solution: The degree of a product is the sum of the degrees. Here, 2 + 3 = 5.
18. What is the Galois group of x^4 - 2 over Q?
a) Z/4Z b) D4 c) S4 d) A4
Answer: b) D4
Solution: The Galois group is isomorphic to the dihedral group D4.
19. How many generators does the cyclic group Z/15Z have?
a) 1 b) 3 c) 8 d) 15
Answer: c) 8
Solution: The number of generators is φ(15), where φ is Euler's totient function. φ(15) = 8.
20. What is the dimension of Q(√2, √3) as a vector space over Q?
a) 2 b) 3 c) 4 d) 6
Answer: c) 4
Solution: A basis is {1, √2, √3, √6}, so the dimension is 4.
21. In the ring Z/6Z, how many units are there?
a) 1 b) 2 c) 3 d) 6
Answer: b) 2
Solution: The units are elements with multiplicative inverses. In Z/6Z, these are 1 and 5.
22. What is the order of the group (Z/2Z × Z/3Z, +)?
a) 3 b) 5 c) 6 d) 12
Answer: c) 6
Solution: The order is the product of the orders of the component groups: 2 * 3 = 6.
23. Which of the following is not a normal subgroup of S4?
a) A4 b) {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} c) {e, (1 2 3 4), (1 3)(2 4), (1 4 3 2)} d) {e, (1 2)}
Answer: d) {e, (1 2)}
Solution: The subgroup {e, (1 2)} is not normal in S4 as it's not closed under conjugation.
24. What is the characteristic of the field C of complex numbers?
a) 0 b) 1 c) 2 d) Infinite
Answer: a) 0
Solution: C has characteristic 0 as no positive integer multiple of 1 equals 0.
25. In the polynomial ring F2[x], what is (x^2 + x + 1)^2?
a) x^4 + x^2 + 1 b) x^4 + x^3 + x^2 + x + 1 c) x^4 + x^2 d) x^4 + 1
Answer: a) x^4 + x^2 + 1
Solution: Expand and simplify, remembering that 1 + 1 = 0 in F2.
26. What is the index of 3Z in Z?
a) 2 b) 3 c) 6 d) Infinite
Answer: b) 3
Solution: The index is the number of cosets, which is |Z/3Z| = 3.
27. Which of the following is not isomorphic to Z/6Z?
a) Z/2Z × Z/3Z b) S3 c) D6 d) C6
Answer: b) S3
Solution: S3 is not abelian, while Z/6Z is abelian.
28. What is the degree of the splitting field of x^3 - 2 over Q?
a) 3 b) 6 c) 9 d) 18
Answer: b) 6
Solution: The splitting field is Q(∛2, ω), where ω is a primitive cube root of unity. [Q(∛2, ω) : Q] = 6.
29. In the ring Z[i], which of the following is irreducible?
a) 2 b) 1 + i c) 2 + 2i d) 5
Answer: b) 1 + i
Solution: 1 + i has prime norm (1^2 + 1^2 = 2) and is thus irreducible.
30. What is the order of GL(2, F2), the general linear group of 2x2 matrices over F2?
a) 4 b) 6 c) 8 d) 16
Answer: b) 6
Solution: There are 6 invertible 2x2 matrices over F2.
31. Which of the following is not a maximal ideal in Z?
a) 2Z b) 3Z c) 5Z d) 6Z
Answer: d) 6Z
Solution: 6Z is not prime, so it's not maximal. The others are maximal as they're generated by
primes.
32. What is the multiplicative order of 2 in Z/7Z?
a) 2 b) 3 c) 6 d) 7
Answer: b) 3
Solution: 2^3 ≡ 1 (mod 7), and this is the smallest positive exponent with this property.
33. How many subgroups does Z/8Z have?
a) 2 b) 3 c) 4 d) 8
Answer: c) 4
Solution: The subgroups correspond to divisors of 8. There are 4 divisors: 1, 2, 4, 8.
34. What is the rank of the free abelian group Z × Z × Z?
a) 1 b) 2 c) 3 d) Infinite
Answer: c) 3
Solution: The rank of a free abelian group is the number of Z factors in its direct product
decomposition.
35. In the ring Z[√-5], which of the following is true?
a) It's a UFD b) It's a PID c) It's Euclidean d) None of the above
Answer: d) None of the above
Solution: Z[√-5] is not a UFD (and thus not a PID or Euclidean) as 6 = 2 * 3 = (1 + √-5)(1 - √-5) are
two different factorizations.
36. What is the Krull dimension of the ring k[x, y], where k is a field?
a) 0 b) 1 c) 2 d) Infinite
Answer: c) 2
Solution: The Krull dimension is the length of the longest chain of prime ideals, which is 2 for k[x,
y].
37. Which of the following groups is not simple?
a) A5 b) PSL(2,7) c) S4 d) Z/5Z
Answer: c) S4
Solution: S4 has a normal subgroup A4, so it's not simple.
38. What is the nilradical of Z/12Z?
a) {0} b) {0, 6} c) {0, 4, 8} d) {0, 3, 6, 9}
Answer: c) {0, 4, 8}
Solution: The nilradical consists of all nilpotent elements. In Z/12Z, these are 0, 4, and 8.
39. What is the degree of the minimal polynomial of i over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 + 1, which has degree 2.
40. In the dihedral group D8, how many elements have order 4?
a) 0 b) 2 c) 4 d) 8
Answer: c) 4
Solution: In D8, there are 4 elements of order 4, namely the rotations by 90° and 270°.
41. What is the Jacobson radical of Z?
a) {0} b) Z c) 2Z d) The set of all prime ideals
Answer: a) {0}
Solution: Z is a commutative ring with 1, so its Jacobson radical is the intersection of all maximal
ideals, which is {0}.
42. Which of the following is not a field extension of Q?
a) Q(√2) b) Q(π) c) Q(i) d) Q[x]
Answer: d) Q[x]
Solution: Q[x] is a ring but not a field, as most polynomials don't have multiplicative inverses.
43. What is the order of the group Aut(Z/15Z)?
a) 4 b) 8 c) 15 d) 16
Answer: b) 8
Solution: Aut(Z/15Z) ≅ (Z/15Z)*, which has order φ(15) = 8.
44. In the ring Z[x], what is the greatest common divisor of x^2 - 1 and x^2 + x - 2?
a) 1 b) x - 1 c) x + 1 d) x^2 - 1
Answer: b) x - 1
Solution: Use the Euclidean algorithm for polynomials to find gcd(x^2 - 1, x^2 + x - 2) = x - 1.
45. What is the number of Sylow 3-subgroups in A4?
a) 1 b) 3 c) 4 d) 12
Answer: c) 4
Solution: By Sylow's theorems, the number of Sylow 3-subgroups must divide 4 and be congruent
to 1 mod 3.
46. Which of the following is a maximal ideal in the ring C[x, y]?
a) (x) b) (x, y) c) (x^2, y) d) (x^2 + y^2 - 1)
Answer: b) (x, y)
Solution: (x, y) is maximal because C[x, y]/(x, y) ≅ C, which is a field.
47. What is the number of conjugacy classes in S4?
a) 4 b) 5 c) 6 d) 24
Answer: b) 5
Solution: The conjugacy classes in S4 correspond to cycle types: (), (12), (123), (1234), (12)(34).
48. In the field F16, what is the multiplicative order of a primitive element?
a) 5 b) 15 c) 16 d) 17
Answer: b) 15
3. Which of the following is not a subgroup of (R, +)?
a) Z b) Q c) {0} d) R*
Answer: d) R*
Solution: R* (non-zero real numbers) is not closed under addition, so it's not a subgroup of (R, +).
4. What is the center of the quaternion group Q8?
a) {1, -1} b) {1, -1, i, -i} c) {1, -1, j, -j} d) {1, -1, k, -k}
Answer: a) {1, -1}
Solution: The center consists of elements that commute with all others. Only 1 and -1 have this
property in Q8.
5. In Z/6Z, what is 4^(-1)?
a) 2 b) 3 c) 4 d) 5
Answer: c) 4
Solution: We need to find x such that 4x ≡ 1 (mod 6). This is satisfied by x = 4.
6. What is the order of GL(2, R), the general linear group of 2x2 matrices over R?
a) Finite b) Countably infinite c) Uncountably infinite d) Zero
Answer: c) Uncountably infinite
Solution: There are uncountably many 2x2 invertible matrices over R.
7. Which of the following is not an ideal in Z?
a) 2Z b) 3Z c) Z d) Z+
Answer: d) Z+
Solution: Z+ (positive integers) is not closed under subtraction, so it's not an ideal.
8. What is the characteristic of the field Z/5Z?
a) 0 b) 2 c) 5 d) Infinite
Answer: c) 5
Solution: The characteristic is the smallest positive integer n such that n·1 = 0 in the field.
9. How many elements are in the quotient group Z/12Z?
a) 6 b) 12 c) 24 d) Infinite
Answer: b) 12
Solution: The order of a quotient group is equal to the order of the original group divided by the
order of the subgroup.
10. What is the degree of the minimal polynomial of √2 over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 - 2, which has degree 2.
11. Which of the following is not a ring?
a) Z b) Q c) R d) C*
Answer: d) C*
Solution: C* (non-zero complex numbers) is not closed under addition, so it's not a ring.
12. What is the order of the alternating group A4?
a) 8 b) 12 c) 24 d) 48
Answer: b) 12
Solution: The order of An is n!/2. For n = 4, this is 4!/2 = 12.
13. In the ring Z[i], what is the norm of 3 + 4i?
a) 5 b) 7 c) 25 d) 49
Answer: c) 25
Solution: The norm of a + bi is a^2 + b^2. Here, 3^2 + 4^2 = 25.
14. What is the number of Sylow 2-subgroups in a group of order 24?
a) 1 b) 3 c) 4 d) 8
Answer: b) 3
Solution: By Sylow's theorems, the number of Sylow 2-subgroups must divide 3 and be congruent
to 1 mod 2.
15. Which of the following is not a field?
a) Q b) R c) C d) Z
Answer: d) Z
Solution: Z (integers) does not have multiplicative inverses for all non-zero elements, so it's not a
field.
16. What is the order of the automorphism group of Z/8Z?
a) 2 b) 4 c) 8 d) 16
Answer: b) 4
Solution: The automorphism group is isomorphic to (Z/8Z)*, which has order 4.
17. In the ring Z[x], what is the degree of (x^2 + 1)(x^3 - 2x + 1)?
a) 3 b) 4 c) 5 d) 6
Answer: c) 5
Solution: The degree of a product is the sum of the degrees. Here, 2 + 3 = 5.
18. What is the Galois group of x^4 - 2 over Q?
a) Z/4Z b) D4 c) S4 d) A4
Answer: b) D4
Solution: The Galois group is isomorphic to the dihedral group D4.
19. How many generators does the cyclic group Z/15Z have?
a) 1 b) 3 c) 8 d) 15
Answer: c) 8
Solution: The number of generators is φ(15), where φ is Euler's totient function. φ(15) = 8.
20. What is the dimension of Q(√2, √3) as a vector space over Q?
a) 2 b) 3 c) 4 d) 6
Answer: c) 4
Solution: A basis is {1, √2, √3, √6}, so the dimension is 4.
21. In the ring Z/6Z, how many units are there?
a) 1 b) 2 c) 3 d) 6
Answer: b) 2
Solution: The units are elements with multiplicative inverses. In Z/6Z, these are 1 and 5.
22. What is the order of the group (Z/2Z × Z/3Z, +)?
a) 3 b) 5 c) 6 d) 12
Answer: c) 6
Solution: The order is the product of the orders of the component groups: 2 * 3 = 6.
23. Which of the following is not a normal subgroup of S4?
a) A4 b) {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} c) {e, (1 2 3 4), (1 3)(2 4), (1 4 3 2)} d) {e, (1 2)}
Answer: d) {e, (1 2)}
Solution: The subgroup {e, (1 2)} is not normal in S4 as it's not closed under conjugation.
24. What is the characteristic of the field C of complex numbers?
a) 0 b) 1 c) 2 d) Infinite
Answer: a) 0
Solution: C has characteristic 0 as no positive integer multiple of 1 equals 0.
25. In the polynomial ring F2[x], what is (x^2 + x + 1)^2?
a) x^4 + x^2 + 1 b) x^4 + x^3 + x^2 + x + 1 c) x^4 + x^2 d) x^4 + 1
Answer: a) x^4 + x^2 + 1
Solution: Expand and simplify, remembering that 1 + 1 = 0 in F2.
26. What is the index of 3Z in Z?
a) 2 b) 3 c) 6 d) Infinite
Answer: b) 3
Solution: The index is the number of cosets, which is |Z/3Z| = 3.
27. Which of the following is not isomorphic to Z/6Z?
a) Z/2Z × Z/3Z b) S3 c) D6 d) C6
Answer: b) S3
Solution: S3 is not abelian, while Z/6Z is abelian.
28. What is the degree of the splitting field of x^3 - 2 over Q?
a) 3 b) 6 c) 9 d) 18
Answer: b) 6
Solution: The splitting field is Q(∛2, ω), where ω is a primitive cube root of unity. [Q(∛2, ω) : Q] = 6.
29. In the ring Z[i], which of the following is irreducible?
a) 2 b) 1 + i c) 2 + 2i d) 5
Answer: b) 1 + i
Solution: 1 + i has prime norm (1^2 + 1^2 = 2) and is thus irreducible.
30. What is the order of GL(2, F2), the general linear group of 2x2 matrices over F2?
a) 4 b) 6 c) 8 d) 16
Answer: b) 6
Solution: There are 6 invertible 2x2 matrices over F2.
31. Which of the following is not a maximal ideal in Z?
a) 2Z b) 3Z c) 5Z d) 6Z
Answer: d) 6Z
Solution: 6Z is not prime, so it's not maximal. The others are maximal as they're generated by
primes.
32. What is the multiplicative order of 2 in Z/7Z?
a) 2 b) 3 c) 6 d) 7
Answer: b) 3
Solution: 2^3 ≡ 1 (mod 7), and this is the smallest positive exponent with this property.
33. How many subgroups does Z/8Z have?
a) 2 b) 3 c) 4 d) 8
Answer: c) 4
Solution: The subgroups correspond to divisors of 8. There are 4 divisors: 1, 2, 4, 8.
34. What is the rank of the free abelian group Z × Z × Z?
a) 1 b) 2 c) 3 d) Infinite
Answer: c) 3
Solution: The rank of a free abelian group is the number of Z factors in its direct product
decomposition.
35. In the ring Z[√-5], which of the following is true?
a) It's a UFD b) It's a PID c) It's Euclidean d) None of the above
Answer: d) None of the above
Solution: Z[√-5] is not a UFD (and thus not a PID or Euclidean) as 6 = 2 * 3 = (1 + √-5)(1 - √-5) are
two different factorizations.
36. What is the Krull dimension of the ring k[x, y], where k is a field?
a) 0 b) 1 c) 2 d) Infinite
Answer: c) 2
Solution: The Krull dimension is the length of the longest chain of prime ideals, which is 2 for k[x,
y].
37. Which of the following groups is not simple?
a) A5 b) PSL(2,7) c) S4 d) Z/5Z
Answer: c) S4
Solution: S4 has a normal subgroup A4, so it's not simple.
38. What is the nilradical of Z/12Z?
a) {0} b) {0, 6} c) {0, 4, 8} d) {0, 3, 6, 9}
Answer: c) {0, 4, 8}
Solution: The nilradical consists of all nilpotent elements. In Z/12Z, these are 0, 4, and 8.
39. What is the degree of the minimal polynomial of i over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 + 1, which has degree 2.
40. In the dihedral group D8, how many elements have order 4?
a) 0 b) 2 c) 4 d) 8
Answer: c) 4
Solution: In D8, there are 4 elements of order 4, namely the rotations by 90° and 270°.
41. What is the Jacobson radical of Z?
a) {0} b) Z c) 2Z d) The set of all prime ideals
Answer: a) {0}
Solution: Z is a commutative ring with 1, so its Jacobson radical is the intersection of all maximal
ideals, which is {0}.
42. Which of the following is not a field extension of Q?
a) Q(√2) b) Q(π) c) Q(i) d) Q[x]
Answer: d) Q[x]
Solution: Q[x] is a ring but not a field, as most polynomials don't have multiplicative inverses.
43. What is the order of the group Aut(Z/15Z)?
a) 4 b) 8 c) 15 d) 16
Answer: b) 8
Solution: Aut(Z/15Z) ≅ (Z/15Z)*, which has order φ(15) = 8.
44. In the ring Z[x], what is the greatest common divisor of x^2 - 1 and x^2 + x - 2?
a) 1 b) x - 1 c) x + 1 d) x^2 - 1
Answer: b) x - 1
Solution: Use the Euclidean algorithm for polynomials to find gcd(x^2 - 1, x^2 + x - 2) = x - 1.
45. What is the number of Sylow 3-subgroups in A4?
a) 1 b) 3 c) 4 d) 12
Answer: c) 4
Solution: By Sylow's theorems, the number of Sylow 3-subgroups must divide 4 and be congruent
to 1 mod 3.
46. Which of the following is a maximal ideal in the ring C[x, y]?
a) (x) b) (x, y) c) (x^2, y) d) (x^2 + y^2 - 1)
Answer: b) (x, y)
Solution: (x, y) is maximal because C[x, y]/(x, y) ≅ C, which is a field.
47. What is the number of conjugacy classes in S4?
a) 4 b) 5 c) 6 d) 24
Answer: b) 5
Solution: The conjugacy classes in S4 correspond to cycle types: (), (12), (123), (1234), (12)(34).
48. In the field F16, what is the multiplicative order of a primitive element?
a) 5 b) 15 c) 16 d) 17
Answer: b) 15
3. Which of the following is not a subgroup of (R, +)?
a) Z b) Q c) {0} d) R*
Answer: d) R*
Solution: R* (non-zero real numbers) is not closed under addition, so it's not a subgroup of (R, +).
4. What is the center of the quaternion group Q8?
a) {1, -1} b) {1, -1, i, -i} c) {1, -1, j, -j} d) {1, -1, k, -k}
Answer: a) {1, -1}
Solution: The center consists of elements that commute with all others. Only 1 and -1 have this
property in Q8.
5. In Z/6Z, what is 4^(-1)?
a) 2 b) 3 c) 4 d) 5
Answer: c) 4
Solution: We need to find x such that 4x ≡ 1 (mod 6). This is satisfied by x = 4.
6. What is the order of GL(2, R), the general linear group of 2x2 matrices over R?
a) Finite b) Countably infinite c) Uncountably infinite d) Zero
Answer: c) Uncountably infinite
Solution: There are uncountably many 2x2 invertible matrices over R.
7. Which of the following is not an ideal in Z?
a) 2Z b) 3Z c) Z d) Z+
Answer: d) Z+
Solution: Z+ (positive integers) is not closed under subtraction, so it's not an ideal.
8. What is the characteristic of the field Z/5Z?
a) 0 b) 2 c) 5 d) Infinite
Answer: c) 5
Solution: The characteristic is the smallest positive integer n such that n·1 = 0 in the field.
9. How many elements are in the quotient group Z/12Z?
a) 6 b) 12 c) 24 d) Infinite
Answer: b) 12
Solution: The order of a quotient group is equal to the order of the original group divided by the
order of the subgroup.
10. What is the degree of the minimal polynomial of √2 over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 - 2, which has degree 2.
11. Which of the following is not a ring?
a) Z b) Q c) R d) C*
Answer: d) C*
Solution: C* (non-zero complex numbers) is not closed under addition, so it's not a ring.
12. What is the order of the alternating group A4?
a) 8 b) 12 c) 24 d) 48
Answer: b) 12
Solution: The order of An is n!/2. For n = 4, this is 4!/2 = 12.
13. In the ring Z[i], what is the norm of 3 + 4i?
a) 5 b) 7 c) 25 d) 49
Answer: c) 25
Solution: The norm of a + bi is a^2 + b^2. Here, 3^2 + 4^2 = 25.
14. What is the number of Sylow 2-subgroups in a group of order 24?
a) 1 b) 3 c) 4 d) 8
Answer: b) 3
Solution: By Sylow's theorems, the number of Sylow 2-subgroups must divide 3 and be congruent
to 1 mod 2.
15. Which of the following is not a field?
a) Q b) R c) C d) Z
Answer: d) Z
Solution: Z (integers) does not have multiplicative inverses for all non-zero elements, so it's not a
field.
16. What is the order of the automorphism group of Z/8Z?
a) 2 b) 4 c) 8 d) 16
Answer: b) 4
Solution: The automorphism group is isomorphic to (Z/8Z)*, which has order 4.
17. In the ring Z[x], what is the degree of (x^2 + 1)(x^3 - 2x + 1)?
a) 3 b) 4 c) 5 d) 6
Answer: c) 5
Solution: The degree of a product is the sum of the degrees. Here, 2 + 3 = 5.
18. What is the Galois group of x^4 - 2 over Q?
a) Z/4Z b) D4 c) S4 d) A4
Answer: b) D4
Solution: The Galois group is isomorphic to the dihedral group D4.
19. How many generators does the cyclic group Z/15Z have?
a) 1 b) 3 c) 8 d) 15
Answer: c) 8
Solution: The number of generators is φ(15), where φ is Euler's totient function. φ(15) = 8.
20. What is the dimension of Q(√2, √3) as a vector space over Q?
a) 2 b) 3 c) 4 d) 6
Answer: c) 4
Solution: A basis is {1, √2, √3, √6}, so the dimension is 4.
21. In the ring Z/6Z, how many units are there?
a) 1 b) 2 c) 3 d) 6
Answer: b) 2
Solution: The units are elements with multiplicative inverses. In Z/6Z, these are 1 and 5.
22. What is the order of the group (Z/2Z × Z/3Z, +)?
a) 3 b) 5 c) 6 d) 12
Answer: c) 6
Solution: The order is the product of the orders of the component groups: 2 * 3 = 6.
23. Which of the following is not a normal subgroup of S4?
a) A4 b) {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} c) {e, (1 2 3 4), (1 3)(2 4), (1 4 3 2)} d) {e, (1 2)}
Answer: d) {e, (1 2)}
Solution: The subgroup {e, (1 2)} is not normal in S4 as it's not closed under conjugation.
24. What is the characteristic of the field C of complex numbers?
a) 0 b) 1 c) 2 d) Infinite
Answer: a) 0
Solution: C has characteristic 0 as no positive integer multiple of 1 equals 0.
25. In the polynomial ring F2[x], what is (x^2 + x + 1)^2?
a) x^4 + x^2 + 1 b) x^4 + x^3 + x^2 + x + 1 c) x^4 + x^2 d) x^4 + 1
Answer: a) x^4 + x^2 + 1
Solution: Expand and simplify, remembering that 1 + 1 = 0 in F2.
26. What is the index of 3Z in Z?
a) 2 b) 3 c) 6 d) Infinite
Answer: b) 3
Solution: The index is the number of cosets, which is |Z/3Z| = 3.
27. Which of the following is not isomorphic to Z/6Z?
a) Z/2Z × Z/3Z b) S3 c) D6 d) C6
Answer: b) S3
Solution: S3 is not abelian, while Z/6Z is abelian.
28. What is the degree of the splitting field of x^3 - 2 over Q?
a) 3 b) 6 c) 9 d) 18
Answer: b) 6
Solution: The splitting field is Q(∛2, ω), where ω is a primitive cube root of unity. [Q(∛2, ω) : Q] = 6.
29. In the ring Z[i], which of the following is irreducible?
a) 2 b) 1 + i c) 2 + 2i d) 5
Answer: b) 1 + i
Solution: 1 + i has prime norm (1^2 + 1^2 = 2) and is thus irreducible.
30. What is the order of GL(2, F2), the general linear group of 2x2 matrices over F2?
a) 4 b) 6 c) 8 d) 16
Answer: b) 6
Solution: There are 6 invertible 2x2 matrices over F2.
31. Which of the following is not a maximal ideal in Z?
a) 2Z b) 3Z c) 5Z d) 6Z
Answer: d) 6Z
Solution: 6Z is not prime, so it's not maximal. The others are maximal as they're generated by
primes.
32. What is the multiplicative order of 2 in Z/7Z?
a) 2 b) 3 c) 6 d) 7
Answer: b) 3
Solution: 2^3 ≡ 1 (mod 7), and this is the smallest positive exponent with this property.
33. How many subgroups does Z/8Z have?
a) 2 b) 3 c) 4 d) 8
Answer: c) 4
Solution: The subgroups correspond to divisors of 8. There are 4 divisors: 1, 2, 4, 8.
34. What is the rank of the free abelian group Z × Z × Z?
a) 1 b) 2 c) 3 d) Infinite
Answer: c) 3
Solution: The rank of a free abelian group is the number of Z factors in its direct product
decomposition.
35. In the ring Z[√-5], which of the following is true?
a) It's a UFD b) It's a PID c) It's Euclidean d) None of the above
Answer: d) None of the above
Solution: Z[√-5] is not a UFD (and thus not a PID or Euclidean) as 6 = 2 * 3 = (1 + √-5)(1 - √-5) are
two different factorizations.
36. What is the Krull dimension of the ring k[x, y], where k is a field?
a) 0 b) 1 c) 2 d) Infinite
Answer: c) 2
Solution: The Krull dimension is the length of the longest chain of prime ideals, which is 2 for k[x,
y].
37. Which of the following groups is not simple?
a) A5 b) PSL(2,7) c) S4 d) Z/5Z
Answer: c) S4
Solution: S4 has a normal subgroup A4, so it's not simple.
38. What is the nilradical of Z/12Z?
a) {0} b) {0, 6} c) {0, 4, 8} d) {0, 3, 6, 9}
Answer: c) {0, 4, 8}
Solution: The nilradical consists of all nilpotent elements. In Z/12Z, these are 0, 4, and 8.
39. What is the degree of the minimal polynomial of i over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 + 1, which has degree 2.
40. In the dihedral group D8, how many elements have order 4?
a) 0 b) 2 c) 4 d) 8
Answer: c) 4
Solution: In D8, there are 4 elements of order 4, namely the rotations by 90° and 270°.
41. What is the Jacobson radical of Z?
a) {0} b) Z c) 2Z d) The set of all prime ideals
Answer: a) {0}
Solution: Z is a commutative ring with 1, so its Jacobson radical is the intersection of all maximal
ideals, which is {0}.
42. Which of the following is not a field extension of Q?
a) Q(√2) b) Q(π) c) Q(i) d) Q[x]
Answer: d) Q[x]
Solution: Q[x] is a ring but not a field, as most polynomials don't have multiplicative inverses.
43. What is the order of the group Aut(Z/15Z)?
a) 4 b) 8 c) 15 d) 16
Answer: b) 8
Solution: Aut(Z/15Z) ≅ (Z/15Z)*, which has order φ(15) = 8.
44. In the ring Z[x], what is the greatest common divisor of x^2 - 1 and x^2 + x - 2?
a) 1 b) x - 1 c) x + 1 d) x^2 - 1
Answer: b) x - 1
Solution: Use the Euclidean algorithm for polynomials to find gcd(x^2 - 1, x^2 + x - 2) = x - 1.
45. What is the number of Sylow 3-subgroups in A4?
a) 1 b) 3 c) 4 d) 12
Answer: c) 4
Solution: By Sylow's theorems, the number of Sylow 3-subgroups must divide 4 and be congruent
to 1 mod 3.
46. Which of the following is a maximal ideal in the ring C[x, y]?
a) (x) b) (x, y) c) (x^2, y) d) (x^2 + y^2 - 1)
Answer: b) (x, y)
Solution: (x, y) is maximal because C[x, y]/(x, y) ≅ C, which is a field.
47. What is the number of conjugacy classes in S4?
a) 4 b) 5 c) 6 d) 24
Answer: b) 5
Solution: The conjugacy classes in S4 correspond to cycle types: (), (12), (123), (1234), (12)(34).
48. In the field F16, what is the multiplicative order of a primitive element?
a) 5 b) 15 c) 16 d) 17
Answer: b) 15
3. Which of the following is not a subgroup of (R, +)?
a) Z b) Q c) {0} d) R*
Answer: d) R*
Solution: R* (non-zero real numbers) is not closed under addition, so it's not a subgroup of (R, +).
4. What is the center of the quaternion group Q8?
a) {1, -1} b) {1, -1, i, -i} c) {1, -1, j, -j} d) {1, -1, k, -k}
Answer: a) {1, -1}
Solution: The center consists of elements that commute with all others. Only 1 and -1 have this
property in Q8.
5. In Z/6Z, what is 4^(-1)?
a) 2 b) 3 c) 4 d) 5
Answer: c) 4
Solution: We need to find x such that 4x ≡ 1 (mod 6). This is satisfied by x = 4.
6. What is the order of GL(2, R), the general linear group of 2x2 matrices over R?
a) Finite b) Countably infinite c) Uncountably infinite d) Zero
Answer: c) Uncountably infinite
Solution: There are uncountably many 2x2 invertible matrices over R.
7. Which of the following is not an ideal in Z?
a) 2Z b) 3Z c) Z d) Z+
Answer: d) Z+
Solution: Z+ (positive integers) is not closed under subtraction, so it's not an ideal.
8. What is the characteristic of the field Z/5Z?
a) 0 b) 2 c) 5 d) Infinite
Answer: c) 5
Solution: The characteristic is the smallest positive integer n such that n·1 = 0 in the field.
9. How many elements are in the quotient group Z/12Z?
a) 6 b) 12 c) 24 d) Infinite
Answer: b) 12
Solution: The order of a quotient group is equal to the order of the original group divided by the
order of the subgroup.
10. What is the degree of the minimal polynomial of √2 over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 - 2, which has degree 2.
11. Which of the following is not a ring?
a) Z b) Q c) R d) C*
Answer: d) C*
Solution: C* (non-zero complex numbers) is not closed under addition, so it's not a ring.
12. What is the order of the alternating group A4?
a) 8 b) 12 c) 24 d) 48
Answer: b) 12
Solution: The order of An is n!/2. For n = 4, this is 4!/2 = 12.
13. In the ring Z[i], what is the norm of 3 + 4i?
a) 5 b) 7 c) 25 d) 49
Answer: c) 25
Solution: The norm of a + bi is a^2 + b^2. Here, 3^2 + 4^2 = 25.
14. What is the number of Sylow 2-subgroups in a group of order 24?
a) 1 b) 3 c) 4 d) 8
Answer: b) 3
Solution: By Sylow's theorems, the number of Sylow 2-subgroups must divide 3 and be congruent
to 1 mod 2.
15. Which of the following is not a field?
a) Q b) R c) C d) Z
Answer: d) Z
Solution: Z (integers) does not have multiplicative inverses for all non-zero elements, so it's not a
field.
16. What is the order of the automorphism group of Z/8Z?
a) 2 b) 4 c) 8 d) 16
Answer: b) 4
Solution: The automorphism group is isomorphic to (Z/8Z)*, which has order 4.
17. In the ring Z[x], what is the degree of (x^2 + 1)(x^3 - 2x + 1)?
a) 3 b) 4 c) 5 d) 6
Answer: c) 5
Solution: The degree of a product is the sum of the degrees. Here, 2 + 3 = 5.
18. What is the Galois group of x^4 - 2 over Q?
a) Z/4Z b) D4 c) S4 d) A4
Answer: b) D4
Solution: The Galois group is isomorphic to the dihedral group D4.
19. How many generators does the cyclic group Z/15Z have?
a) 1 b) 3 c) 8 d) 15
Answer: c) 8
Solution: The number of generators is φ(15), where φ is Euler's totient function. φ(15) = 8.
20. What is the dimension of Q(√2, √3) as a vector space over Q?
a) 2 b) 3 c) 4 d) 6
Answer: c) 4
Solution: A basis is {1, √2, √3, √6}, so the dimension is 4.
21. In the ring Z/6Z, how many units are there?
a) 1 b) 2 c) 3 d) 6
Answer: b) 2
Solution: The units are elements with multiplicative inverses. In Z/6Z, these are 1 and 5.
22. What is the order of the group (Z/2Z × Z/3Z, +)?
a) 3 b) 5 c) 6 d) 12
Answer: c) 6
Solution: The order is the product of the orders of the component groups: 2 * 3 = 6.
23. Which of the following is not a normal subgroup of S4?
a) A4 b) {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} c) {e, (1 2 3 4), (1 3)(2 4), (1 4 3 2)} d) {e, (1 2)}
Answer: d) {e, (1 2)}
Solution: The subgroup {e, (1 2)} is not normal in S4 as it's not closed under conjugation.
24. What is the characteristic of the field C of complex numbers?
a) 0 b) 1 c) 2 d) Infinite
Answer: a) 0
Solution: C has characteristic 0 as no positive integer multiple of 1 equals 0.
25. In the polynomial ring F2[x], what is (x^2 + x + 1)^2?
a) x^4 + x^2 + 1 b) x^4 + x^3 + x^2 + x + 1 c) x^4 + x^2 d) x^4 + 1
Answer: a) x^4 + x^2 + 1
Solution: Expand and simplify, remembering that 1 + 1 = 0 in F2.
26. What is the index of 3Z in Z?
a) 2 b) 3 c) 6 d) Infinite
Answer: b) 3
Solution: The index is the number of cosets, which is |Z/3Z| = 3.
27. Which of the following is not isomorphic to Z/6Z?
a) Z/2Z × Z/3Z b) S3 c) D6 d) C6
Answer: b) S3
Solution: S3 is not abelian, while Z/6Z is abelian.
28. What is the degree of the splitting field of x^3 - 2 over Q?
a) 3 b) 6 c) 9 d) 18
Answer: b) 6
Solution: The splitting field is Q(∛2, ω), where ω is a primitive cube root of unity. [Q(∛2, ω) : Q] = 6.
29. In the ring Z[i], which of the following is irreducible?
a) 2 b) 1 + i c) 2 + 2i d) 5
Answer: b) 1 + i
Solution: 1 + i has prime norm (1^2 + 1^2 = 2) and is thus irreducible.
30. What is the order of GL(2, F2), the general linear group of 2x2 matrices over F2?
a) 4 b) 6 c) 8 d) 16
Answer: b) 6
Solution: There are 6 invertible 2x2 matrices over F2.
31. Which of the following is not a maximal ideal in Z?
a) 2Z b) 3Z c) 5Z d) 6Z
Answer: d) 6Z
Solution: 6Z is not prime, so it's not maximal. The others are maximal as they're generated by
primes.
32. What is the multiplicative order of 2 in Z/7Z?
a) 2 b) 3 c) 6 d) 7
Answer: b) 3
Solution: 2^3 ≡ 1 (mod 7), and this is the smallest positive exponent with this property.
33. How many subgroups does Z/8Z have?
a) 2 b) 3 c) 4 d) 8
Answer: c) 4
Solution: The subgroups correspond to divisors of 8. There are 4 divisors: 1, 2, 4, 8.
34. What is the rank of the free abelian group Z × Z × Z?
a) 1 b) 2 c) 3 d) Infinite
Answer: c) 3
Solution: The rank of a free abelian group is the number of Z factors in its direct product
decomposition.
35. In the ring Z[√-5], which of the following is true?
a) It's a UFD b) It's a PID c) It's Euclidean d) None of the above
Answer: d) None of the above
Solution: Z[√-5] is not a UFD (and thus not a PID or Euclidean) as 6 = 2 * 3 = (1 + √-5)(1 - √-5) are
two different factorizations.
36. What is the Krull dimension of the ring k[x, y], where k is a field?
a) 0 b) 1 c) 2 d) Infinite
Answer: c) 2
Solution: The Krull dimension is the length of the longest chain of prime ideals, which is 2 for k[x,
y].
37. Which of the following groups is not simple?
a) A5 b) PSL(2,7) c) S4 d) Z/5Z
Answer: c) S4
Solution: S4 has a normal subgroup A4, so it's not simple.
38. What is the nilradical of Z/12Z?
a) {0} b) {0, 6} c) {0, 4, 8} d) {0, 3, 6, 9}
Answer: c) {0, 4, 8}
Solution: The nilradical consists of all nilpotent elements. In Z/12Z, these are 0, 4, and 8.
39. What is the degree of the minimal polynomial of i over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 + 1, which has degree 2.
40. In the dihedral group D8, how many elements have order 4?
a) 0 b) 2 c) 4 d) 8
Answer: c) 4
Solution: In D8, there are 4 elements of order 4, namely the rotations by 90° and 270°.
41. What is the Jacobson radical of Z?
a) {0} b) Z c) 2Z d) The set of all prime ideals
Answer: a) {0}
Solution: Z is a commutative ring with 1, so its Jacobson radical is the intersection of all maximal
ideals, which is {0}.
42. Which of the following is not a field extension of Q?
a) Q(√2) b) Q(π) c) Q(i) d) Q[x]
Answer: d) Q[x]
Solution: Q[x] is a ring but not a field, as most polynomials don't have multiplicative inverses.
43. What is the order of the group Aut(Z/15Z)?
a) 4 b) 8 c) 15 d) 16
Answer: b) 8
Solution: Aut(Z/15Z) ≅ (Z/15Z)*, which has order φ(15) = 8.
44. In the ring Z[x], what is the greatest common divisor of x^2 - 1 and x^2 + x - 2?
a) 1 b) x - 1 c) x + 1 d) x^2 - 1
Answer: b) x - 1
Solution: Use the Euclidean algorithm for polynomials to find gcd(x^2 - 1, x^2 + x - 2) = x - 1.
45. What is the number of Sylow 3-subgroups in A4?
a) 1 b) 3 c) 4 d) 12
Answer: c) 4
Solution: By Sylow's theorems, the number of Sylow 3-subgroups must divide 4 and be congruent
to 1 mod 3.
46. Which of the following is a maximal ideal in the ring C[x, y]?
a) (x) b) (x, y) c) (x^2, y) d) (x^2 + y^2 - 1)
Answer: b) (x, y)
Solution: (x, y) is maximal because C[x, y]/(x, y) ≅ C, which is a field.
47. What is the number of conjugacy classes in S4?
a) 4 b) 5 c) 6 d) 24
Answer: b) 5
Solution: The conjugacy classes in S4 correspond to cycle types: (), (12), (123), (1234), (12)(34).
48. In the field F16, what is the multiplicative order of a primitive element?
a) 5 b) 15 c) 16 d) 17
Answer: b) 15
3. Which of the following is not a subgroup of (R, +)?
a) Z b) Q c) {0} d) R*
Answer: d) R*
Solution: R* (non-zero real numbers) is not closed under addition, so it's not a subgroup of (R, +).
4. What is the center of the quaternion group Q8?
a) {1, -1} b) {1, -1, i, -i} c) {1, -1, j, -j} d) {1, -1, k, -k}
Answer: a) {1, -1}
Solution: The center consists of elements that commute with all others. Only 1 and -1 have this
property in Q8.
5. In Z/6Z, what is 4^(-1)?
a) 2 b) 3 c) 4 d) 5
Answer: c) 4
Solution: We need to find x such that 4x ≡ 1 (mod 6). This is satisfied by x = 4.
6. What is the order of GL(2, R), the general linear group of 2x2 matrices over R?
a) Finite b) Countably infinite c) Uncountably infinite d) Zero
Answer: c) Uncountably infinite
Solution: There are uncountably many 2x2 invertible matrices over R.
7. Which of the following is not an ideal in Z?
a) 2Z b) 3Z c) Z d) Z+
Answer: d) Z+
Solution: Z+ (positive integers) is not closed under subtraction, so it's not an ideal.
8. What is the characteristic of the field Z/5Z?
a) 0 b) 2 c) 5 d) Infinite
Answer: c) 5
Solution: The characteristic is the smallest positive integer n such that n·1 = 0 in the field.
9. How many elements are in the quotient group Z/12Z?
a) 6 b) 12 c) 24 d) Infinite
Answer: b) 12
Solution: The order of a quotient group is equal to the order of the original group divided by the
order of the subgroup.
10. What is the degree of the minimal polynomial of √2 over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 - 2, which has degree 2.
11. Which of the following is not a ring?
a) Z b) Q c) R d) C*
Answer: d) C*
Solution: C* (non-zero complex numbers) is not closed under addition, so it's not a ring.
12. What is the order of the alternating group A4?
a) 8 b) 12 c) 24 d) 48
Answer: b) 12
Solution: The order of An is n!/2. For n = 4, this is 4!/2 = 12.
13. In the ring Z[i], what is the norm of 3 + 4i?
a) 5 b) 7 c) 25 d) 49
Answer: c) 25
Solution: The norm of a + bi is a^2 + b^2. Here, 3^2 + 4^2 = 25.
14. What is the number of Sylow 2-subgroups in a group of order 24?
a) 1 b) 3 c) 4 d) 8
Answer: b) 3
Solution: By Sylow's theorems, the number of Sylow 2-subgroups must divide 3 and be congruent
to 1 mod 2.
15. Which of the following is not a field?
a) Q b) R c) C d) Z
Answer: d) Z
Solution: Z (integers) does not have multiplicative inverses for all non-zero elements, so it's not a
field.
16. What is the order of the automorphism group of Z/8Z?
a) 2 b) 4 c) 8 d) 16
Answer: b) 4
Solution: The automorphism group is isomorphic to (Z/8Z)*, which has order 4.
17. In the ring Z[x], what is the degree of (x^2 + 1)(x^3 - 2x + 1)?
a) 3 b) 4 c) 5 d) 6
Answer: c) 5
Solution: The degree of a product is the sum of the degrees. Here, 2 + 3 = 5.
18. What is the Galois group of x^4 - 2 over Q?
a) Z/4Z b) D4 c) S4 d) A4
Answer: b) D4
Solution: The Galois group is isomorphic to the dihedral group D4.
19. How many generators does the cyclic group Z/15Z have?
a) 1 b) 3 c) 8 d) 15
Answer: c) 8
Solution: The number of generators is φ(15), where φ is Euler's totient function. φ(15) = 8.
20. What is the dimension of Q(√2, √3) as a vector space over Q?
a) 2 b) 3 c) 4 d) 6
Answer: c) 4
Solution: A basis is {1, √2, √3, √6}, so the dimension is 4.
21. In the ring Z/6Z, how many units are there?
a) 1 b) 2 c) 3 d) 6
Answer: b) 2
Solution: The units are elements with multiplicative inverses. In Z/6Z, these are 1 and 5.
22. What is the order of the group (Z/2Z × Z/3Z, +)?
a) 3 b) 5 c) 6 d) 12
Answer: c) 6
Solution: The order is the product of the orders of the component groups: 2 * 3 = 6.
23. Which of the following is not a normal subgroup of S4?
a) A4 b) {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} c) {e, (1 2 3 4), (1 3)(2 4), (1 4 3 2)} d) {e, (1 2)}
Answer: d) {e, (1 2)}
Solution: The subgroup {e, (1 2)} is not normal in S4 as it's not closed under conjugation.
24. What is the characteristic of the field C of complex numbers?
a) 0 b) 1 c) 2 d) Infinite
Answer: a) 0
Solution: C has characteristic 0 as no positive integer multiple of 1 equals 0.
25. In the polynomial ring F2[x], what is (x^2 + x + 1)^2?
a) x^4 + x^2 + 1 b) x^4 + x^3 + x^2 + x + 1 c) x^4 + x^2 d) x^4 + 1
Answer: a) x^4 + x^2 + 1
Solution: Expand and simplify, remembering that 1 + 1 = 0 in F2.
26. What is the index of 3Z in Z?
a) 2 b) 3 c) 6 d) Infinite
Answer: b) 3
Solution: The index is the number of cosets, which is |Z/3Z| = 3.
27. Which of the following is not isomorphic to Z/6Z?
a) Z/2Z × Z/3Z b) S3 c) D6 d) C6
Answer: b) S3
Solution: S3 is not abelian, while Z/6Z is abelian.
28. What is the degree of the splitting field of x^3 - 2 over Q?
a) 3 b) 6 c) 9 d) 18
Answer: b) 6
Solution: The splitting field is Q(∛2, ω), where ω is a primitive cube root of unity. [Q(∛2, ω) : Q] = 6.
29. In the ring Z[i], which of the following is irreducible?
a) 2 b) 1 + i c) 2 + 2i d) 5
Answer: b) 1 + i
Solution: 1 + i has prime norm (1^2 + 1^2 = 2) and is thus irreducible.
30. What is the order of GL(2, F2), the general linear group of 2x2 matrices over F2?
a) 4 b) 6 c) 8 d) 16
Answer: b) 6
Solution: There are 6 invertible 2x2 matrices over F2.
31. Which of the following is not a maximal ideal in Z?
a) 2Z b) 3Z c) 5Z d) 6Z
Answer: d) 6Z
Solution: 6Z is not prime, so it's not maximal. The others are maximal as they're generated by
primes.
32. What is the multiplicative order of 2 in Z/7Z?
a) 2 b) 3 c) 6 d) 7
Answer: b) 3
Solution: 2^3 ≡ 1 (mod 7), and this is the smallest positive exponent with this property.
33. How many subgroups does Z/8Z have?
a) 2 b) 3 c) 4 d) 8
Answer: c) 4
Solution: The subgroups correspond to divisors of 8. There are 4 divisors: 1, 2, 4, 8.
34. What is the rank of the free abelian group Z × Z × Z?
a) 1 b) 2 c) 3 d) Infinite
Answer: c) 3
Solution: The rank of a free abelian group is the number of Z factors in its direct product
decomposition.
35. In the ring Z[√-5], which of the following is true?
a) It's a UFD b) It's a PID c) It's Euclidean d) None of the above
Answer: d) None of the above
Solution: Z[√-5] is not a UFD (and thus not a PID or Euclidean) as 6 = 2 * 3 = (1 + √-5)(1 - √-5) are
two different factorizations.
36. What is the Krull dimension of the ring k[x, y], where k is a field?
a) 0 b) 1 c) 2 d) Infinite
Answer: c) 2
Solution: The Krull dimension is the length of the longest chain of prime ideals, which is 2 for k[x,
y].
37. Which of the following groups is not simple?
a) A5 b) PSL(2,7) c) S4 d) Z/5Z
Answer: c) S4
Solution: S4 has a normal subgroup A4, so it's not simple.
38. What is the nilradical of Z/12Z?
a) {0} b) {0, 6} c) {0, 4, 8} d) {0, 3, 6, 9}
Answer: c) {0, 4, 8}
Solution: The nilradical consists of all nilpotent elements. In Z/12Z, these are 0, 4, and 8.
39. What is the degree of the minimal polynomial of i over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 + 1, which has degree 2.
40. In the dihedral group D8, how many elements have order 4?
a) 0 b) 2 c) 4 d) 8
Answer: c) 4
Solution: In D8, there are 4 elements of order 4, namely the rotations by 90° and 270°.
41. What is the Jacobson radical of Z?
a) {0} b) Z c) 2Z d) The set of all prime ideals
Answer: a) {0}
Solution: Z is a commutative ring with 1, so its Jacobson radical is the intersection of all maximal
ideals, which is {0}.
42. Which of the following is not a field extension of Q?
a) Q(√2) b) Q(π) c) Q(i) d) Q[x]
Answer: d) Q[x]
Solution: Q[x] is a ring but not a field, as most polynomials don't have multiplicative inverses.
43. What is the order of the group Aut(Z/15Z)?
a) 4 b) 8 c) 15 d) 16
Answer: b) 8
Solution: Aut(Z/15Z) ≅ (Z/15Z)*, which has order φ(15) = 8.
44. In the ring Z[x], what is the greatest common divisor of x^2 - 1 and x^2 + x - 2?
a) 1 b) x - 1 c) x + 1 d) x^2 - 1
Answer: b) x - 1
Solution: Use the Euclidean algorithm for polynomials to find gcd(x^2 - 1, x^2 + x - 2) = x - 1.
45. What is the number of Sylow 3-subgroups in A4?
a) 1 b) 3 c) 4 d) 12
Answer: c) 4
Solution: By Sylow's theorems, the number of Sylow 3-subgroups must divide 4 and be congruent
to 1 mod 3.
46. Which of the following is a maximal ideal in the ring C[x, y]?
a) (x) b) (x, y) c) (x^2, y) d) (x^2 + y^2 - 1)
Answer: b) (x, y)
Solution: (x, y) is maximal because C[x, y]/(x, y) ≅ C, which is a field.
47. What is the number of conjugacy classes in S4?
a) 4 b) 5 c) 6 d) 24
Answer: b) 5
Solution: The conjugacy classes in S4 correspond to cycle types: (), (12), (123), (1234), (12)(34).
48. In the field F16, what is the multiplicative order of a primitive element?
a) 5 b) 15 c) 16 d) 17
Answer: b) 15
3. Which of the following is not a subgroup of (R, +)?
a) Z b) Q c) {0} d) R*
Answer: d) R*
Solution: R* (non-zero real numbers) is not closed under addition, so it's not a subgroup of (R, +).
4. What is the center of the quaternion group Q8?
a) {1, -1} b) {1, -1, i, -i} c) {1, -1, j, -j} d) {1, -1, k, -k}
Answer: a) {1, -1}
Solution: The center consists of elements that commute with all others. Only 1 and -1 have this
property in Q8.
5. In Z/6Z, what is 4^(-1)?
a) 2 b) 3 c) 4 d) 5
Answer: c) 4
Solution: We need to find x such that 4x ≡ 1 (mod 6). This is satisfied by x = 4.
6. What is the order of GL(2, R), the general linear group of 2x2 matrices over R?
a) Finite b) Countably infinite c) Uncountably infinite d) Zero
Answer: c) Uncountably infinite
Solution: There are uncountably many 2x2 invertible matrices over R.
7. Which of the following is not an ideal in Z?
a) 2Z b) 3Z c) Z d) Z+
Answer: d) Z+
Solution: Z+ (positive integers) is not closed under subtraction, so it's not an ideal.
8. What is the characteristic of the field Z/5Z?
a) 0 b) 2 c) 5 d) Infinite
Answer: c) 5
Solution: The characteristic is the smallest positive integer n such that n·1 = 0 in the field.
9. How many elements are in the quotient group Z/12Z?
a) 6 b) 12 c) 24 d) Infinite
Answer: b) 12
Solution: The order of a quotient group is equal to the order of the original group divided by the
order of the subgroup.
10. What is the degree of the minimal polynomial of √2 over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 - 2, which has degree 2.
11. Which of the following is not a ring?
a) Z b) Q c) R d) C*
Answer: d) C*
Solution: C* (non-zero complex numbers) is not closed under addition, so it's not a ring.
12. What is the order of the alternating group A4?
a) 8 b) 12 c) 24 d) 48
Answer: b) 12
Solution: The order of An is n!/2. For n = 4, this is 4!/2 = 12.
13. In the ring Z[i], what is the norm of 3 + 4i?
a) 5 b) 7 c) 25 d) 49
Answer: c) 25
Solution: The norm of a + bi is a^2 + b^2. Here, 3^2 + 4^2 = 25.
14. What is the number of Sylow 2-subgroups in a group of order 24?
a) 1 b) 3 c) 4 d) 8
Answer: b) 3
Solution: By Sylow's theorems, the number of Sylow 2-subgroups must divide 3 and be congruent
to 1 mod 2.
15. Which of the following is not a field?
a) Q b) R c) C d) Z
Answer: d) Z
Solution: Z (integers) does not have multiplicative inverses for all non-zero elements, so it's not a
field.
16. What is the order of the automorphism group of Z/8Z?
a) 2 b) 4 c) 8 d) 16
Answer: b) 4
Solution: The automorphism group is isomorphic to (Z/8Z)*, which has order 4.
17. In the ring Z[x], what is the degree of (x^2 + 1)(x^3 - 2x + 1)?
a) 3 b) 4 c) 5 d) 6
Answer: c) 5
Solution: The degree of a product is the sum of the degrees. Here, 2 + 3 = 5.
18. What is the Galois group of x^4 - 2 over Q?
a) Z/4Z b) D4 c) S4 d) A4
Answer: b) D4
Solution: The Galois group is isomorphic to the dihedral group D4.
19. How many generators does the cyclic group Z/15Z have?
a) 1 b) 3 c) 8 d) 15
Answer: c) 8
Solution: The number of generators is φ(15), where φ is Euler's totient function. φ(15) = 8.
20. What is the dimension of Q(√2, √3) as a vector space over Q?
a) 2 b) 3 c) 4 d) 6
Answer: c) 4
Solution: A basis is {1, √2, √3, √6}, so the dimension is 4.
21. In the ring Z/6Z, how many units are there?
a) 1 b) 2 c) 3 d) 6
Answer: b) 2
Solution: The units are elements with multiplicative inverses. In Z/6Z, these are 1 and 5.
22. What is the order of the group (Z/2Z × Z/3Z, +)?
a) 3 b) 5 c) 6 d) 12
Answer: c) 6
Solution: The order is the product of the orders of the component groups: 2 * 3 = 6.
23. Which of the following is not a normal subgroup of S4?
a) A4 b) {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} c) {e, (1 2 3 4), (1 3)(2 4), (1 4 3 2)} d) {e, (1 2)}
Answer: d) {e, (1 2)}
Solution: The subgroup {e, (1 2)} is not normal in S4 as it's not closed under conjugation.
24. What is the characteristic of the field C of complex numbers?
a) 0 b) 1 c) 2 d) Infinite
Answer: a) 0
Solution: C has characteristic 0 as no positive integer multiple of 1 equals 0.
25. In the polynomial ring F2[x], what is (x^2 + x + 1)^2?
a) x^4 + x^2 + 1 b) x^4 + x^3 + x^2 + x + 1 c) x^4 + x^2 d) x^4 + 1
Answer: a) x^4 + x^2 + 1
Solution: Expand and simplify, remembering that 1 + 1 = 0 in F2.
26. What is the index of 3Z in Z?
a) 2 b) 3 c) 6 d) Infinite
Answer: b) 3
Solution: The index is the number of cosets, which is |Z/3Z| = 3.
27. Which of the following is not isomorphic to Z/6Z?
a) Z/2Z × Z/3Z b) S3 c) D6 d) C6
Answer: b) S3
Solution: S3 is not abelian, while Z/6Z is abelian.
28. What is the degree of the splitting field of x^3 - 2 over Q?
a) 3 b) 6 c) 9 d) 18
Answer: b) 6
Solution: The splitting field is Q(∛2, ω), where ω is a primitive cube root of unity. [Q(∛2, ω) : Q] = 6.
29. In the ring Z[i], which of the following is irreducible?
a) 2 b) 1 + i c) 2 + 2i d) 5
Answer: b) 1 + i
Solution: 1 + i has prime norm (1^2 + 1^2 = 2) and is thus irreducible.
30. What is the order of GL(2, F2), the general linear group of 2x2 matrices over F2?
a) 4 b) 6 c) 8 d) 16
Answer: b) 6
Solution: There are 6 invertible 2x2 matrices over F2.
31. Which of the following is not a maximal ideal in Z?
a) 2Z b) 3Z c) 5Z d) 6Z
Answer: d) 6Z
Solution: 6Z is not prime, so it's not maximal. The others are maximal as they're generated by
primes.
32. What is the multiplicative order of 2 in Z/7Z?
a) 2 b) 3 c) 6 d) 7
Answer: b) 3
Solution: 2^3 ≡ 1 (mod 7), and this is the smallest positive exponent with this property.
33. How many subgroups does Z/8Z have?
a) 2 b) 3 c) 4 d) 8
Answer: c) 4
Solution: The subgroups correspond to divisors of 8. There are 4 divisors: 1, 2, 4, 8.
34. What is the rank of the free abelian group Z × Z × Z?
a) 1 b) 2 c) 3 d) Infinite
Answer: c) 3
Solution: The rank of a free abelian group is the number of Z factors in its direct product
decomposition.
35. In the ring Z[√-5], which of the following is true?
a) It's a UFD b) It's a PID c) It's Euclidean d) None of the above
Answer: d) None of the above
Solution: Z[√-5] is not a UFD (and thus not a PID or Euclidean) as 6 = 2 * 3 = (1 + √-5)(1 - √-5) are
two different factorizations.
36. What is the Krull dimension of the ring k[x, y], where k is a field?
a) 0 b) 1 c) 2 d) Infinite
Answer: c) 2
Solution: The Krull dimension is the length of the longest chain of prime ideals, which is 2 for k[x,
y].
37. Which of the following groups is not simple?
a) A5 b) PSL(2,7) c) S4 d) Z/5Z
Answer: c) S4
Solution: S4 has a normal subgroup A4, so it's not simple.
38. What is the nilradical of Z/12Z?
a) {0} b) {0, 6} c) {0, 4, 8} d) {0, 3, 6, 9}
Answer: c) {0, 4, 8}
Solution: The nilradical consists of all nilpotent elements. In Z/12Z, these are 0, 4, and 8.
39. What is the degree of the minimal polynomial of i over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 + 1, which has degree 2.
40. In the dihedral group D8, how many elements have order 4?
a) 0 b) 2 c) 4 d) 8
Answer: c) 4
Solution: In D8, there are 4 elements of order 4, namely the rotations by 90° and 270°.
41. What is the Jacobson radical of Z?
a) {0} b) Z c) 2Z d) The set of all prime ideals
Answer: a) {0}
Solution: Z is a commutative ring with 1, so its Jacobson radical is the intersection of all maximal
ideals, which is {0}.
42. Which of the following is not a field extension of Q?
a) Q(√2) b) Q(π) c) Q(i) d) Q[x]
Answer: d) Q[x]
Solution: Q[x] is a ring but not a field, as most polynomials don't have multiplicative inverses.
43. What is the order of the group Aut(Z/15Z)?
a) 4 b) 8 c) 15 d) 16
Answer: b) 8
Solution: Aut(Z/15Z) ≅ (Z/15Z)*, which has order φ(15) = 8.
44. In the ring Z[x], what is the greatest common divisor of x^2 - 1 and x^2 + x - 2?
a) 1 b) x - 1 c) x + 1 d) x^2 - 1
Answer: b) x - 1
Solution: Use the Euclidean algorithm for polynomials to find gcd(x^2 - 1, x^2 + x - 2) = x - 1.
45. What is the number of Sylow 3-subgroups in A4?
a) 1 b) 3 c) 4 d) 12
Answer: c) 4
Solution: By Sylow's theorems, the number of Sylow 3-subgroups must divide 4 and be congruent
to 1 mod 3.
46. Which of the following is a maximal ideal in the ring C[x, y]?
a) (x) b) (x, y) c) (x^2, y) d) (x^2 + y^2 - 1)
Answer: b) (x, y)
Solution: (x, y) is maximal because C[x, y]/(x, y) ≅ C, which is a field.
47. What is the number of conjugacy classes in S4?
a) 4 b) 5 c) 6 d) 24
Answer: b) 5
Solution: The conjugacy classes in S4 correspond to cycle types: (), (12), (123), (1234), (12)(34).
48. In the field F16, what is the multiplicative order of a primitive element?
a) 5 b) 15 c) 16 d) 17
Answer: b) 15
3. Which of the following is not a subgroup of (R, +)?
a) Z b) Q c) {0} d) R*
Answer: d) R*
Solution: R* (non-zero real numbers) is not closed under addition, so it's not a subgroup of (R, +).
4. What is the center of the quaternion group Q8?
a) {1, -1} b) {1, -1, i, -i} c) {1, -1, j, -j} d) {1, -1, k, -k}
Answer: a) {1, -1}
Solution: The center consists of elements that commute with all others. Only 1 and -1 have this
property in Q8.
5. In Z/6Z, what is 4^(-1)?
a) 2 b) 3 c) 4 d) 5
Answer: c) 4
Solution: We need to find x such that 4x ≡ 1 (mod 6). This is satisfied by x = 4.
6. What is the order of GL(2, R), the general linear group of 2x2 matrices over R?
a) Finite b) Countably infinite c) Uncountably infinite d) Zero
Answer: c) Uncountably infinite
Solution: There are uncountably many 2x2 invertible matrices over R.
7. Which of the following is not an ideal in Z?
a) 2Z b) 3Z c) Z d) Z+
Answer: d) Z+
Solution: Z+ (positive integers) is not closed under subtraction, so it's not an ideal.
8. What is the characteristic of the field Z/5Z?
a) 0 b) 2 c) 5 d) Infinite
Answer: c) 5
Solution: The characteristic is the smallest positive integer n such that n·1 = 0 in the field.
9. How many elements are in the quotient group Z/12Z?
a) 6 b) 12 c) 24 d) Infinite
Answer: b) 12
Solution: The order of a quotient group is equal to the order of the original group divided by the
order of the subgroup.
10. What is the degree of the minimal polynomial of √2 over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 - 2, which has degree 2.
11. Which of the following is not a ring?
a) Z b) Q c) R d) C*
Answer: d) C*
Solution: C* (non-zero complex numbers) is not closed under addition, so it's not a ring.
12. What is the order of the alternating group A4?
a) 8 b) 12 c) 24 d) 48
Answer: b) 12
Solution: The order of An is n!/2. For n = 4, this is 4!/2 = 12.
13. In the ring Z[i], what is the norm of 3 + 4i?
a) 5 b) 7 c) 25 d) 49
Answer: c) 25
Solution: The norm of a + bi is a^2 + b^2. Here, 3^2 + 4^2 = 25.
14. What is the number of Sylow 2-subgroups in a group of order 24?
a) 1 b) 3 c) 4 d) 8
Answer: b) 3
Solution: By Sylow's theorems, the number of Sylow 2-subgroups must divide 3 and be congruent
to 1 mod 2.
15. Which of the following is not a field?
a) Q b) R c) C d) Z
Answer: d) Z
Solution: Z (integers) does not have multiplicative inverses for all non-zero elements, so it's not a
field.
16. What is the order of the automorphism group of Z/8Z?
a) 2 b) 4 c) 8 d) 16
Answer: b) 4
Solution: The automorphism group is isomorphic to (Z/8Z)*, which has order 4.
17. In the ring Z[x], what is the degree of (x^2 + 1)(x^3 - 2x + 1)?
a) 3 b) 4 c) 5 d) 6
Answer: c) 5
Solution: The degree of a product is the sum of the degrees. Here, 2 + 3 = 5.
18. What is the Galois group of x^4 - 2 over Q?
a) Z/4Z b) D4 c) S4 d) A4
Answer: b) D4
Solution: The Galois group is isomorphic to the dihedral group D4.
19. How many generators does the cyclic group Z/15Z have?
a) 1 b) 3 c) 8 d) 15
Answer: c) 8
Solution: The number of generators is φ(15), where φ is Euler's totient function. φ(15) = 8.
20. What is the dimension of Q(√2, √3) as a vector space over Q?
a) 2 b) 3 c) 4 d) 6
Answer: c) 4
Solution: A basis is {1, √2, √3, √6}, so the dimension is 4.
21. In the ring Z/6Z, how many units are there?
a) 1 b) 2 c) 3 d) 6
Answer: b) 2
Solution: The units are elements with multiplicative inverses. In Z/6Z, these are 1 and 5.
22. What is the order of the group (Z/2Z × Z/3Z, +)?
a) 3 b) 5 c) 6 d) 12
Answer: c) 6
Solution: The order is the product of the orders of the component groups: 2 * 3 = 6.
23. Which of the following is not a normal subgroup of S4?
a) A4 b) {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} c) {e, (1 2 3 4), (1 3)(2 4), (1 4 3 2)} d) {e, (1 2)}
Answer: d) {e, (1 2)}
Solution: The subgroup {e, (1 2)} is not normal in S4 as it's not closed under conjugation.
24. What is the characteristic of the field C of complex numbers?
a) 0 b) 1 c) 2 d) Infinite
Answer: a) 0
Solution: C has characteristic 0 as no positive integer multiple of 1 equals 0.
25. In the polynomial ring F2[x], what is (x^2 + x + 1)^2?
a) x^4 + x^2 + 1 b) x^4 + x^3 + x^2 + x + 1 c) x^4 + x^2 d) x^4 + 1
Answer: a) x^4 + x^2 + 1
Solution: Expand and simplify, remembering that 1 + 1 = 0 in F2.
26. What is the index of 3Z in Z?
a) 2 b) 3 c) 6 d) Infinite
Answer: b) 3
Solution: The index is the number of cosets, which is |Z/3Z| = 3.
27. Which of the following is not isomorphic to Z/6Z?
a) Z/2Z × Z/3Z b) S3 c) D6 d) C6
Answer: b) S3
Solution: S3 is not abelian, while Z/6Z is abelian.
28. What is the degree of the splitting field of x^3 - 2 over Q?
a) 3 b) 6 c) 9 d) 18
Answer: b) 6
Solution: The splitting field is Q(∛2, ω), where ω is a primitive cube root of unity. [Q(∛2, ω) : Q] = 6.
29. In the ring Z[i], which of the following is irreducible?
a) 2 b) 1 + i c) 2 + 2i d) 5
Answer: b) 1 + i
Solution: 1 + i has prime norm (1^2 + 1^2 = 2) and is thus irreducible.
30. What is the order of GL(2, F2), the general linear group of 2x2 matrices over F2?
a) 4 b) 6 c) 8 d) 16
Answer: b) 6
Solution: There are 6 invertible 2x2 matrices over F2.
31. Which of the following is not a maximal ideal in Z?
a) 2Z b) 3Z c) 5Z d) 6Z
Answer: d) 6Z
Solution: 6Z is not prime, so it's not maximal. The others are maximal as they're generated by
primes.
32. What is the multiplicative order of 2 in Z/7Z?
a) 2 b) 3 c) 6 d) 7
Answer: b) 3
Solution: 2^3 ≡ 1 (mod 7), and this is the smallest positive exponent with this property.
33. How many subgroups does Z/8Z have?
a) 2 b) 3 c) 4 d) 8
Answer: c) 4
Solution: The subgroups correspond to divisors of 8. There are 4 divisors: 1, 2, 4, 8.
34. What is the rank of the free abelian group Z × Z × Z?
a) 1 b) 2 c) 3 d) Infinite
Answer: c) 3
Solution: The rank of a free abelian group is the number of Z factors in its direct product
decomposition.
35. In the ring Z[√-5], which of the following is true?
a) It's a UFD b) It's a PID c) It's Euclidean d) None of the above
Answer: d) None of the above
Solution: Z[√-5] is not a UFD (and thus not a PID or Euclidean) as 6 = 2 * 3 = (1 + √-5)(1 - √-5) are
two different factorizations.
36. What is the Krull dimension of the ring k[x, y], where k is a field?
a) 0 b) 1 c) 2 d) Infinite
Answer: c) 2
Solution: The Krull dimension is the length of the longest chain of prime ideals, which is 2 for k[x,
y].
37. Which of the following groups is not simple?
a) A5 b) PSL(2,7) c) S4 d) Z/5Z
Answer: c) S4
Solution: S4 has a normal subgroup A4, so it's not simple.
38. What is the nilradical of Z/12Z?
a) {0} b) {0, 6} c) {0, 4, 8} d) {0, 3, 6, 9}
Answer: c) {0, 4, 8}
Solution: The nilradical consists of all nilpotent elements. In Z/12Z, these are 0, 4, and 8.
39. What is the degree of the minimal polynomial of i over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 + 1, which has degree 2.
40. In the dihedral group D8, how many elements have order 4?
a) 0 b) 2 c) 4 d) 8
Answer: c) 4
Solution: In D8, there are 4 elements of order 4, namely the rotations by 90° and 270°.
41. What is the Jacobson radical of Z?
a) {0} b) Z c) 2Z d) The set of all prime ideals
Answer: a) {0}
Solution: Z is a commutative ring with 1, so its Jacobson radical is the intersection of all maximal
ideals, which is {0}.
42. Which of the following is not a field extension of Q?
a) Q(√2) b) Q(π) c) Q(i) d) Q[x]
Answer: d) Q[x]
Solution: Q[x] is a ring but not a field, as most polynomials don't have multiplicative inverses.
43. What is the order of the group Aut(Z/15Z)?
a) 4 b) 8 c) 15 d) 16
Answer: b) 8
Solution: Aut(Z/15Z) ≅ (Z/15Z)*, which has order φ(15) = 8.
44. In the ring Z[x], what is the greatest common divisor of x^2 - 1 and x^2 + x - 2?
a) 1 b) x - 1 c) x + 1 d) x^2 - 1
Answer: b) x - 1
Solution: Use the Euclidean algorithm for polynomials to find gcd(x^2 - 1, x^2 + x - 2) = x - 1.
45. What is the number of Sylow 3-subgroups in A4?
a) 1 b) 3 c) 4 d) 12
Answer: c) 4
Solution: By Sylow's theorems, the number of Sylow 3-subgroups must divide 4 and be congruent
to 1 mod 3.
46. Which of the following is a maximal ideal in the ring C[x, y]?
a) (x) b) (x, y) c) (x^2, y) d) (x^2 + y^2 - 1)
Answer: b) (x, y)
Solution: (x, y) is maximal because C[x, y]/(x, y) ≅ C, which is a field.
47. What is the number of conjugacy classes in S4?
a) 4 b) 5 c) 6 d) 24
Answer: b) 5
Solution: The conjugacy classes in S4 correspond to cycle types: (), (12), (123), (1234), (12)(34).
48. In the field F16, what is the multiplicative order of a primitive element?
a) 5 b) 15 c) 16 d) 17
Answer: b) 15
3. Which of the following is not a subgroup of (R, +)?
a) Z b) Q c) {0} d) R*
Answer: d) R*
Solution: R* (non-zero real numbers) is not closed under addition, so it's not a subgroup of (R, +).
4. What is the center of the quaternion group Q8?
a) {1, -1} b) {1, -1, i, -i} c) {1, -1, j, -j} d) {1, -1, k, -k}
Answer: a) {1, -1}
Solution: The center consists of elements that commute with all others. Only 1 and -1 have this
property in Q8.
5. In Z/6Z, what is 4^(-1)?
a) 2 b) 3 c) 4 d) 5
Answer: c) 4
Solution: We need to find x such that 4x ≡ 1 (mod 6). This is satisfied by x = 4.
6. What is the order of GL(2, R), the general linear group of 2x2 matrices over R?
a) Finite b) Countably infinite c) Uncountably infinite d) Zero
Answer: c) Uncountably infinite
Solution: There are uncountably many 2x2 invertible matrices over R.
7. Which of the following is not an ideal in Z?
a) 2Z b) 3Z c) Z d) Z+
Answer: d) Z+
Solution: Z+ (positive integers) is not closed under subtraction, so it's not an ideal.
8. What is the characteristic of the field Z/5Z?
a) 0 b) 2 c) 5 d) Infinite
Answer: c) 5
Solution: The characteristic is the smallest positive integer n such that n·1 = 0 in the field.
9. How many elements are in the quotient group Z/12Z?
a) 6 b) 12 c) 24 d) Infinite
Answer: b) 12
Solution: The order of a quotient group is equal to the order of the original group divided by the
order of the subgroup.
10. What is the degree of the minimal polynomial of √2 over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 - 2, which has degree 2.
11. Which of the following is not a ring?
a) Z b) Q c) R d) C*
Answer: d) C*
Solution: C* (non-zero complex numbers) is not closed under addition, so it's not a ring.
12. What is the order of the alternating group A4?
a) 8 b) 12 c) 24 d) 48
Answer: b) 12
Solution: The order of An is n!/2. For n = 4, this is 4!/2 = 12.
13. In the ring Z[i], what is the norm of 3 + 4i?
a) 5 b) 7 c) 25 d) 49
Answer: c) 25
Solution: The norm of a + bi is a^2 + b^2. Here, 3^2 + 4^2 = 25.
14. What is the number of Sylow 2-subgroups in a group of order 24?
a) 1 b) 3 c) 4 d) 8
Answer: b) 3
Solution: By Sylow's theorems, the number of Sylow 2-subgroups must divide 3 and be congruent
to 1 mod 2.
15. Which of the following is not a field?
a) Q b) R c) C d) Z
Answer: d) Z
Solution: Z (integers) does not have multiplicative inverses for all non-zero elements, so it's not a
field.
16. What is the order of the automorphism group of Z/8Z?
a) 2 b) 4 c) 8 d) 16
Answer: b) 4
Solution: The automorphism group is isomorphic to (Z/8Z)*, which has order 4.
17. In the ring Z[x], what is the degree of (x^2 + 1)(x^3 - 2x + 1)?
a) 3 b) 4 c) 5 d) 6
Answer: c) 5
Solution: The degree of a product is the sum of the degrees. Here, 2 + 3 = 5.
18. What is the Galois group of x^4 - 2 over Q?
a) Z/4Z b) D4 c) S4 d) A4
Answer: b) D4
Solution: The Galois group is isomorphic to the dihedral group D4.
19. How many generators does the cyclic group Z/15Z have?
a) 1 b) 3 c) 8 d) 15
Answer: c) 8
Solution: The number of generators is φ(15), where φ is Euler's totient function. φ(15) = 8.
20. What is the dimension of Q(√2, √3) as a vector space over Q?
a) 2 b) 3 c) 4 d) 6
Answer: c) 4
Solution: A basis is {1, √2, √3, √6}, so the dimension is 4.
21. In the ring Z/6Z, how many units are there?
a) 1 b) 2 c) 3 d) 6
Answer: b) 2
Solution: The units are elements with multiplicative inverses. In Z/6Z, these are 1 and 5.
22. What is the order of the group (Z/2Z × Z/3Z, +)?
a) 3 b) 5 c) 6 d) 12
Answer: c) 6
Solution: The order is the product of the orders of the component groups: 2 * 3 = 6.
23. Which of the following is not a normal subgroup of S4?
a) A4 b) {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} c) {e, (1 2 3 4), (1 3)(2 4), (1 4 3 2)} d) {e, (1 2)}
Answer: d) {e, (1 2)}
Solution: The subgroup {e, (1 2)} is not normal in S4 as it's not closed under conjugation.
24. What is the characteristic of the field C of complex numbers?
a) 0 b) 1 c) 2 d) Infinite
Answer: a) 0
Solution: C has characteristic 0 as no positive integer multiple of 1 equals 0.
25. In the polynomial ring F2[x], what is (x^2 + x + 1)^2?
a) x^4 + x^2 + 1 b) x^4 + x^3 + x^2 + x + 1 c) x^4 + x^2 d) x^4 + 1
Answer: a) x^4 + x^2 + 1
Solution: Expand and simplify, remembering that 1 + 1 = 0 in F2.
26. What is the index of 3Z in Z?
a) 2 b) 3 c) 6 d) Infinite
Answer: b) 3
Solution: The index is the number of cosets, which is |Z/3Z| = 3.
27. Which of the following is not isomorphic to Z/6Z?
a) Z/2Z × Z/3Z b) S3 c) D6 d) C6
Answer: b) S3
Solution: S3 is not abelian, while Z/6Z is abelian.
28. What is the degree of the splitting field of x^3 - 2 over Q?
a) 3 b) 6 c) 9 d) 18
Answer: b) 6
Solution: The splitting field is Q(∛2, ω), where ω is a primitive cube root of unity. [Q(∛2, ω) : Q] = 6.
29. In the ring Z[i], which of the following is irreducible?
a) 2 b) 1 + i c) 2 + 2i d) 5
Answer: b) 1 + i
Solution: 1 + i has prime norm (1^2 + 1^2 = 2) and is thus irreducible.
30. What is the order of GL(2, F2), the general linear group of 2x2 matrices over F2?
a) 4 b) 6 c) 8 d) 16
Answer: b) 6
Solution: There are 6 invertible 2x2 matrices over F2.
31. Which of the following is not a maximal ideal in Z?
a) 2Z b) 3Z c) 5Z d) 6Z
Answer: d) 6Z
Solution: 6Z is not prime, so it's not maximal. The others are maximal as they're generated by
primes.
32. What is the multiplicative order of 2 in Z/7Z?
a) 2 b) 3 c) 6 d) 7
Answer: b) 3
Solution: 2^3 ≡ 1 (mod 7), and this is the smallest positive exponent with this property.
33. How many subgroups does Z/8Z have?
a) 2 b) 3 c) 4 d) 8
Answer: c) 4
Solution: The subgroups correspond to divisors of 8. There are 4 divisors: 1, 2, 4, 8.
34. What is the rank of the free abelian group Z × Z × Z?
a) 1 b) 2 c) 3 d) Infinite
Answer: c) 3
Solution: The rank of a free abelian group is the number of Z factors in its direct product
decomposition.
35. In the ring Z[√-5], which of the following is true?
a) It's a UFD b) It's a PID c) It's Euclidean d) None of the above
Answer: d) None of the above
Solution: Z[√-5] is not a UFD (and thus not a PID or Euclidean) as 6 = 2 * 3 = (1 + √-5)(1 - √-5) are
two different factorizations.
36. What is the Krull dimension of the ring k[x, y], where k is a field?
a) 0 b) 1 c) 2 d) Infinite
Answer: c) 2
Solution: The Krull dimension is the length of the longest chain of prime ideals, which is 2 for k[x,
y].
37. Which of the following groups is not simple?
a) A5 b) PSL(2,7) c) S4 d) Z/5Z
Answer: c) S4
Solution: S4 has a normal subgroup A4, so it's not simple.
38. What is the nilradical of Z/12Z?
a) {0} b) {0, 6} c) {0, 4, 8} d) {0, 3, 6, 9}
Answer: c) {0, 4, 8}
Solution: The nilradical consists of all nilpotent elements. In Z/12Z, these are 0, 4, and 8.
39. What is the degree of the minimal polynomial of i over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 + 1, which has degree 2.
40. In the dihedral group D8, how many elements have order 4?
a) 0 b) 2 c) 4 d) 8
Answer: c) 4
Solution: In D8, there are 4 elements of order 4, namely the rotations by 90° and 270°.
41. What is the Jacobson radical of Z?
a) {0} b) Z c) 2Z d) The set of all prime ideals
Answer: a) {0}
Solution: Z is a commutative ring with 1, so its Jacobson radical is the intersection of all maximal
ideals, which is {0}.
42. Which of the following is not a field extension of Q?
a) Q(√2) b) Q(π) c) Q(i) d) Q[x]
Answer: d) Q[x]
Solution: Q[x] is a ring but not a field, as most polynomials don't have multiplicative inverses.
43. What is the order of the group Aut(Z/15Z)?
a) 4 b) 8 c) 15 d) 16
Answer: b) 8
Solution: Aut(Z/15Z) ≅ (Z/15Z)*, which has order φ(15) = 8.
44. In the ring Z[x], what is the greatest common divisor of x^2 - 1 and x^2 + x - 2?
a) 1 b) x - 1 c) x + 1 d) x^2 - 1
Answer: b) x - 1
Solution: Use the Euclidean algorithm for polynomials to find gcd(x^2 - 1, x^2 + x - 2) = x - 1.
45. What is the number of Sylow 3-subgroups in A4?
a) 1 b) 3 c) 4 d) 12
Answer: c) 4
Solution: By Sylow's theorems, the number of Sylow 3-subgroups must divide 4 and be congruent
to 1 mod 3.
46. Which of the following is a maximal ideal in the ring C[x, y]?
a) (x) b) (x, y) c) (x^2, y) d) (x^2 + y^2 - 1)
Answer: b) (x, y)
Solution: (x, y) is maximal because C[x, y]/(x, y) ≅ C, which is a field.
47. What is the number of conjugacy classes in S4?
a) 4 b) 5 c) 6 d) 24
Answer: b) 5
Solution: The conjugacy classes in S4 correspond to cycle types: (), (12), (123), (1234), (12)(34).
48. In the field F16, what is the multiplicative order of a primitive element?
a) 5 b) 15 c) 16 d) 17
Answer: b) 15
3. Which of the following is not a subgroup of (R, +)?
a) Z b) Q c) {0} d) R*
Answer: d) R*
Solution: R* (non-zero real numbers) is not closed under addition, so it's not a subgroup of (R, +).
4. What is the center of the quaternion group Q8?
a) {1, -1} b) {1, -1, i, -i} c) {1, -1, j, -j} d) {1, -1, k, -k}
Answer: a) {1, -1}
Solution: The center consists of elements that commute with all others. Only 1 and -1 have this
property in Q8.
5. In Z/6Z, what is 4^(-1)?
a) 2 b) 3 c) 4 d) 5
Answer: c) 4
Solution: We need to find x such that 4x ≡ 1 (mod 6). This is satisfied by x = 4.
6. What is the order of GL(2, R), the general linear group of 2x2 matrices over R?
a) Finite b) Countably infinite c) Uncountably infinite d) Zero
Answer: c) Uncountably infinite
Solution: There are uncountably many 2x2 invertible matrices over R.
7. Which of the following is not an ideal in Z?
a) 2Z b) 3Z c) Z d) Z+
Answer: d) Z+
Solution: Z+ (positive integers) is not closed under subtraction, so it's not an ideal.
8. What is the characteristic of the field Z/5Z?
a) 0 b) 2 c) 5 d) Infinite
Answer: c) 5
Solution: The characteristic is the smallest positive integer n such that n·1 = 0 in the field.
9. How many elements are in the quotient group Z/12Z?
a) 6 b) 12 c) 24 d) Infinite
Answer: b) 12
Solution: The order of a quotient group is equal to the order of the original group divided by the
order of the subgroup.
10. What is the degree of the minimal polynomial of √2 over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 - 2, which has degree 2.
11. Which of the following is not a ring?
a) Z b) Q c) R d) C*
Answer: d) C*
Solution: C* (non-zero complex numbers) is not closed under addition, so it's not a ring.
12. What is the order of the alternating group A4?
a) 8 b) 12 c) 24 d) 48
Answer: b) 12
Solution: The order of An is n!/2. For n = 4, this is 4!/2 = 12.
13. In the ring Z[i], what is the norm of 3 + 4i?
a) 5 b) 7 c) 25 d) 49
Answer: c) 25
Solution: The norm of a + bi is a^2 + b^2. Here, 3^2 + 4^2 = 25.
14. What is the number of Sylow 2-subgroups in a group of order 24?
a) 1 b) 3 c) 4 d) 8
Answer: b) 3
Solution: By Sylow's theorems, the number of Sylow 2-subgroups must divide 3 and be congruent
to 1 mod 2.
15. Which of the following is not a field?
a) Q b) R c) C d) Z
Answer: d) Z
Solution: Z (integers) does not have multiplicative inverses for all non-zero elements, so it's not a
field.
16. What is the order of the automorphism group of Z/8Z?
a) 2 b) 4 c) 8 d) 16
Answer: b) 4
Solution: The automorphism group is isomorphic to (Z/8Z)*, which has order 4.
17. In the ring Z[x], what is the degree of (x^2 + 1)(x^3 - 2x + 1)?
a) 3 b) 4 c) 5 d) 6
Answer: c) 5
Solution: The degree of a product is the sum of the degrees. Here, 2 + 3 = 5.
18. What is the Galois group of x^4 - 2 over Q?
a) Z/4Z b) D4 c) S4 d) A4
Answer: b) D4
Solution: The Galois group is isomorphic to the dihedral group D4.
19. How many generators does the cyclic group Z/15Z have?
a) 1 b) 3 c) 8 d) 15
Answer: c) 8
Solution: The number of generators is φ(15), where φ is Euler's totient function. φ(15) = 8.
20. What is the dimension of Q(√2, √3) as a vector space over Q?
a) 2 b) 3 c) 4 d) 6
Answer: c) 4
Solution: A basis is {1, √2, √3, √6}, so the dimension is 4.
21. In the ring Z/6Z, how many units are there?
a) 1 b) 2 c) 3 d) 6
Answer: b) 2
Solution: The units are elements with multiplicative inverses. In Z/6Z, these are 1 and 5.
22. What is the order of the group (Z/2Z × Z/3Z, +)?
a) 3 b) 5 c) 6 d) 12
Answer: c) 6
Solution: The order is the product of the orders of the component groups: 2 * 3 = 6.
23. Which of the following is not a normal subgroup of S4?
a) A4 b) {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} c) {e, (1 2 3 4), (1 3)(2 4), (1 4 3 2)} d) {e, (1 2)}
Answer: d) {e, (1 2)}
Solution: The subgroup {e, (1 2)} is not normal in S4 as it's not closed under conjugation.
24. What is the characteristic of the field C of complex numbers?
a) 0 b) 1 c) 2 d) Infinite
Answer: a) 0
Solution: C has characteristic 0 as no positive integer multiple of 1 equals 0.
25. In the polynomial ring F2[x], what is (x^2 + x + 1)^2?
a) x^4 + x^2 + 1 b) x^4 + x^3 + x^2 + x + 1 c) x^4 + x^2 d) x^4 + 1
Answer: a) x^4 + x^2 + 1
Solution: Expand and simplify, remembering that 1 + 1 = 0 in F2.
26. What is the index of 3Z in Z?
a) 2 b) 3 c) 6 d) Infinite
Answer: b) 3
Solution: The index is the number of cosets, which is |Z/3Z| = 3.
27. Which of the following is not isomorphic to Z/6Z?
a) Z/2Z × Z/3Z b) S3 c) D6 d) C6
Answer: b) S3
Solution: S3 is not abelian, while Z/6Z is abelian.
28. What is the degree of the splitting field of x^3 - 2 over Q?
a) 3 b) 6 c) 9 d) 18
Answer: b) 6
Solution: The splitting field is Q(∛2, ω), where ω is a primitive cube root of unity. [Q(∛2, ω) : Q] = 6.
29. In the ring Z[i], which of the following is irreducible?
a) 2 b) 1 + i c) 2 + 2i d) 5
Answer: b) 1 + i
Solution: 1 + i has prime norm (1^2 + 1^2 = 2) and is thus irreducible.
30. What is the order of GL(2, F2), the general linear group of 2x2 matrices over F2?
a) 4 b) 6 c) 8 d) 16
Answer: b) 6
Solution: There are 6 invertible 2x2 matrices over F2.
31. Which of the following is not a maximal ideal in Z?
a) 2Z b) 3Z c) 5Z d) 6Z
Answer: d) 6Z
Solution: 6Z is not prime, so it's not maximal. The others are maximal as they're generated by
primes.
32. What is the multiplicative order of 2 in Z/7Z?
a) 2 b) 3 c) 6 d) 7
Answer: b) 3
Solution: 2^3 ≡ 1 (mod 7), and this is the smallest positive exponent with this property.
33. How many subgroups does Z/8Z have?
a) 2 b) 3 c) 4 d) 8
Answer: c) 4
Solution: The subgroups correspond to divisors of 8. There are 4 divisors: 1, 2, 4, 8.
34. What is the rank of the free abelian group Z × Z × Z?
a) 1 b) 2 c) 3 d) Infinite
Answer: c) 3
Solution: The rank of a free abelian group is the number of Z factors in its direct product
decomposition.
35. In the ring Z[√-5], which of the following is true?
a) It's a UFD b) It's a PID c) It's Euclidean d) None of the above
Answer: d) None of the above
Solution: Z[√-5] is not a UFD (and thus not a PID or Euclidean) as 6 = 2 * 3 = (1 + √-5)(1 - √-5) are
two different factorizations.
36. What is the Krull dimension of the ring k[x, y], where k is a field?
a) 0 b) 1 c) 2 d) Infinite
Answer: c) 2
Solution: The Krull dimension is the length of the longest chain of prime ideals, which is 2 for k[x,
y].
37. Which of the following groups is not simple?
a) A5 b) PSL(2,7) c) S4 d) Z/5Z
Answer: c) S4
Solution: S4 has a normal subgroup A4, so it's not simple.
38. What is the nilradical of Z/12Z?
a) {0} b) {0, 6} c) {0, 4, 8} d) {0, 3, 6, 9}
Answer: c) {0, 4, 8}
Solution: The nilradical consists of all nilpotent elements. In Z/12Z, these are 0, 4, and 8.
39. What is the degree of the minimal polynomial of i over Q?
a) 1 b) 2 c) 3 d) 4
Answer: b) 2
Solution: The minimal polynomial is x^2 + 1, which has degree 2.
40. In the dihedral group D8, how many elements have order 4?
a) 0 b) 2 c) 4 d) 8
Answer: c) 4
Solution: In D8, there are 4 elements of order 4, namely the rotations by 90° and 270°.
41. What is the Jacobson radical of Z?
a) {0} b) Z c) 2Z d) The set of all prime ideals
Answer: a) {0}
Solution: Z is a commutative ring with 1, so its Jacobson radical is the intersection of all maximal
ideals, which is {0}.
42. Which of the following is not a field extension of Q?
a) Q(√2) b) Q(π) c) Q(i) d) Q[x]
Answer: d) Q[x]
Solution: Q[x] is a ring but not a field, as most polynomials don't have multiplicative inverses.
43. What is the order of the group Aut(Z/15Z)?
a) 4 b) 8 c) 15 d) 16
Answer: b) 8
Solution: Aut(Z/15Z) ≅ (Z/15Z)*, which has order φ(15) = 8.
44. In the ring Z[x], what is the greatest common divisor of x^2 - 1 and x^2 + x - 2?
a) 1 b) x - 1 c) x + 1 d) x^2 - 1
Answer: b) x - 1
Solution: Use the Euclidean algorithm for polynomials to find gcd(x^2 - 1, x^2 + x - 2) = x - 1.
45. What is the number of Sylow 3-subgroups in A4?
a) 1 b) 3 c) 4 d) 12
Answer: c) 4
Solution: By Sylow's theorems, the number of Sylow 3-subgroups must divide 4 and be congruent
to 1 mod 3.
46. Which of the following is a maximal ideal in the ring C[x, y]?
a) (x) b) (x, y) c) (x^2, y) d) (x^2 + y^2 - 1)
Answer: b) (x, y)
Solution: (x, y) is maximal because C[x, y]/(x, y) ≅ C, which is a field.
47. What is the number of conjugacy classes in S4?
a) 4 b) 5 c) 6 d) 24
Answer: b) 5
Solution: The conjugacy classes in S4 correspond to cycle types: (), (12), (123), (1234), (12)(34).
48. In the field F16, what is the multiplicative order of a primitive element?
a) 5 b) 15 c) 16 d) 17
Answer: b) 15
Solution: A primitive
