Abstract Algebra II: Master Group, Ring, and Field Theory Concepts

Question 1

In ring theory, what is the key connection between homomorphisms and isomorphisms?

Accepted Answer

Isomorphisms are a specific type of homomorphism that maintain all ring operations.

Answer

Homomorphisms always lead to isomorphisms between the rings involved.

Answer

Homomorphisms and isomorphisms are unrelated concepts in ring theory.

Answer

Every ring is isomorphic to one of its subrings.

Question 2

In the direct product group (G 	imes H), where G and H are finite groups with orders (m) and (n), respectively, what is the order of the direct product group (G 	imes H)?

Accepted Answer

The order of (G 	imes H) is (m 	times n).

Answer

The order of (G 	imes H) is (m + n).

Answer

The order of (G 	imes H) is the greatest common divisor of (m) and (n).

Answer

The order of (G 	imes H) is the least common multiple of (m) and (n).

Question 3

Construct the finite field with 16 elements. Determine its characteristic and identify its prime subfield.

Accepted Answer

GF(16) has a characteristic of 2 and a prime subfield of GF(2).

Answer

GF(16) has a characteristic of 4 and a prime subfield of GF(4).

Answer

GF(16) has a characteristic of 8 and a prime subfield of GF(8).

Answer

GF(16) has a characteristic of 16 and a prime subfield of GF(16).

Question 4

In public-key cryptography like RSA, how are cyclic groups used?

Accepted Answer

Serving as encryption and decryption keys, with group operations ensuring secure key combination

Answer

Generating large prime numbers for public and private keys

Answer

Constructing one-way functions for public-key cryptography security

Answer

Generating random numbers for session keys in public-key cryptography

Question 5

Determine if the polynomial ring (mathbb{Z}[x]) constitutes a Euclidean domain. Provide clear definitions and logical proofs to support your conclusion.

Accepted Answer

(b) (mathbb{Z}[x]) is not a Euclidean domain as it lacks the division algorithm property.

Answer

(a) (mathbb{Z}[x]) is a Euclidean domain due to its nature as a principal ideal domain.

Answer

(c) (mathbb{Z}[x]) qualifies as a Euclidean domain because it adheres to the unique factorization domain condition.

Question 6

Define field extensions and explain their significance in abstract algebra, providing specific examples and applications.

Accepted Answer

Field extensions are new fields created by adding elements to an existing field, inheriting its operations and properties.

Accepted Answer

Field extensions are crucial in Galois theory, offering insights into polynomial equation symmetries and solvability.

Accepted Answer

Field extensions are used in algebraic geometry to construct varieties and curves with specific properties.

Answer

Field extensions have limited applications outside abstract algebra.

Question 7

Which statement accurately describes the symmetric group S(n)?

Accepted Answer

The number of conjugacy classes in S(n) equals the number of distinct cycle types in S(n).

Accepted Answer

S(n) is isomorphic to the group of n x n permutation matrices.

Answer

S(n) is always a solvable group.

Answer

The center of S(n) is always a trivial group.

Question 8

Which of the following sets form a finite group under the specified operation?

Accepted Answer

Permutations of 4 elements

Answer

Integers under addition

Answer

Positive real numbers under multiplication

Answer

Complex numbers under subtraction

Question 9

Which of the following best describes a Galois field?

Accepted Answer

A finite field with a prime number of elements

Answer

A finite field of odd order

Answer

A field that is a subfield of a larger field

Answer

A field that is algebraically closed

Question 10

A key implication of the Jordan-Hölder theorem is:

Accepted Answer

Every finite group possesses a composition series

Answer

Composition series for solvable groups are unique

Answer

Semisimple groups always have a composition series

Answer

Every abelian group has a composition series

Question 11

A defining characteristic of a division ring is:

Accepted Answer

Absence of zero divisors

Answer

Commutativity under the ring operations

Answer

Existence of an identity element under addition

Answer

Being a field

Question 12

What is the characteristic of the finite field GF(p)?

Accepted Answer

p

Answer

0

Answer

1

Answer

p^2

Question 13

Which of the following is a homomorphic image of the additive group of integers Z?

Accepted Answer

The group of integers modulo 3 under addition

Answer

The group of real numbers under addition

Answer

The group of complex numbers under multiplication

Answer

The group of permutations on 3 elements

Question 14

Which of the following is a subring of the field of complex numbers?

Accepted Answer

The set of all real numbers

Answer

The set of all rational numbers

Answer

The set of all integers

Answer

The set of all quaternions

Question 15

In the symmetric group S_3, which of the following is NOT a subgroup?

Accepted Answer

{(), (12), (13), (23), (132)}

Answer

{(), (12)}

Answer

{(), (12), (13)}

Answer

{(), (12), (13), (23), (123)}

Question 16

Which of the following is a field?

Accepted Answer

Q

Answer

Z_4

Answer

Z_6

Answer

Z

Question 17

What is the characteristic of the field GF(p)?

Accepted Answer

p

Answer

2

Answer

0

Answer

1

Question 18

What is the number of generators of the cyclic group Z_n?

Accepted Answer

1

Answer

n - 1

Answer

n

Answer

n + 1

Question 19

Consider the ring Z[x]. Which of the following is an ideal of Z[x]?

Accepted Answer

The set of all polynomials in Z[x] divisible by x.

Answer

The set of all polynomials in Z[x] with even degree.

Answer

The set of all polynomials in Z[x] with rational coefficients.

Answer

The set of all polynomials in Z[x] with constant term 0.

Question 20

Which of the following is a field?

Accepted Answer

The set of all polynomials over a field.

Answer

The set of all rational numbers.

Answer

The set of all integers.

Answer

The set of all real numbers greater than 0.

Question 21

Consider the group G = {1, -1, i, -i}. What is the order of the group G?

Accepted Answer

4

Answer

2

Answer

8

Answer

16

Question 22

Which of the following is a consequence of the Fundamental Theorem of Algebra?

Accepted Answer

Every polynomial of degree n has exactly n roots in the complex numbers.

Answer

Every polynomial with real coefficients has at least one real root.

Answer

Every polynomial with rational coefficients has at least one rational root.

Answer

Every polynomial with integer coefficients has at least one integer root.

Question 23

Consider the field F = GF(7). What is the number of elements in the field F?

Accepted Answer

7

Answer

2

Answer

4

Answer

14

Question 24

What is the group of automorphisms of the group Z/12Z?

Accepted Answer

Z/11Z

Answer

Z/12Z

Answer

Z/4Z

Answer

Z/6Z

Question 25

What is the characteristic of the field GF(p^n)?

Accepted Answer

p

Answer

0

Answer

n

Answer

p^n

Question 26

Which among the options is a homomorphic image of the group Z×Z?

Accepted Answer

Z/2Z×Z/2Z

Answer

Z

Answer

S3

Answer

D4

Question 27

Determine the characteristic of the finite field F_p.

Accepted Answer

p

Answer

0

Answer

1

Answer

p^2

Question 28

Give the definition of a Noetherian ring. Prove that a ring R is Noetherian if and only if every ideal in R is finitely generated.

Accepted Answer

A ring R is Noetherian if every ideal in R is finitely generated.

Answer

A ring R is Noetherian if every ascending chain of ideals in R is finite.

Answer

A ring R is Noetherian if it has a maximal ideal.

Answer

A ring R is Noetherian if it is a principal ideal ring.

Question 29

What is the cardinality of the ring Z_8?

Accepted Answer

8

Answer

2

Answer

4

Answer

16

Question 30

Which set does NOT satisfy the definition of a field?

Accepted Answer

The set of quaternions with non-zero norm

Answer

The set of rational numbers

Answer

The set of integers modulo 5

Answer

The set of complex numbers

Question 31

What is the center of the group D_8?

Accepted Answer

C_2

Answer

C_4

Answer

C_8

Answer

D_4

Question 32

Which of the following sets is NOT a subgroup of the symmetric group S3?

Accepted Answer

The set of permutations that map 1 to 3

Answer

The set of even permutations

Answer

The set of permutations that fix the element 1

Answer

The set of permutations that map 2 to 3

Question 33

Which of the following is a homomorphic image of the group Z under the operation of addition?

Accepted Answer

The group of integers modulo 5 under the operation of addition

Answer

The group Z under the operation of multiplication

Answer

The group of rational numbers under the operation of addition

Answer

The group of real numbers under the operation of addition

Question 34

Which of the following is a simple group?

Accepted Answer

The alternating group A5

Answer

The cyclic group of order 4

Answer

The dihedral group of order 8

Answer

The symmetric group S4

Question 35

Which of the following is a minimal polynomial of the element 2 + sqrt(5) over the field Q?

Accepted Answer

x^2 - 2x - 3

Answer

x^2 - 2

Answer

x^2 - 4

Answer

x^2 - 4x + 2

Question 36

Which of the following is a characteristic subgroup of the group GL(n,R)?

Accepted Answer

The set of all matrices with determinant 1

Answer

The set of all matrices with trace 0

Answer

The set of all matrices with eigenvalues equal to 1

Answer

The set of all matrices with rank n

Question 37

A group is nilpotent if and only if its lower central series terminates at:

Accepted Answer

The trivial subgroup.

Answer

The center of the group.

Answer

The commutator subgroup of the group.

Answer

The group itself.

Question 38

Which ring is not an integral domain?

Accepted Answer

The ring of continuous functions on the closed unit interval.

Answer

The ring of integers.

Answer

The ring of polynomials over a field.

Answer

The ring of matrices over a field.

Question 39

Which characteristic is equivalent to an integral domain definition?

Accepted Answer

There are no zero divisors.

Answer

All non-zero elements possess multiplicative inverses.

Answer

The only divisors of 0 are 0 and 1.

Answer

The structure is a field.

Question 40

What key feature characterizes a simple group?

Accepted Answer

Absence of proper normal subgroups

Answer

Cyclic structure

Answer

Finite order

Answer

Solvable structure

Question 41

What fundamental requirement must a ring fulfill to become a vector space over a field?

Accepted Answer

Addition and scalar multiplication operations

Answer

Field structure

Answer

Integral domain structure

Answer

Additive group structure.

Question 42

In a ring with unity, which element is nilpotent?

Accepted Answer

An element a such that a^n = 0 for some positive integer n

Answer

An element a such that a commutes with every element in the ring

Answer

An element a such that a^-1 does not exist

Answer

An element a such that a is a unit

Question 43

What is the order of a finite field?

Accepted Answer

A prime number p or a power of a prime p^n, where n is a positive integer

Answer

Arbitrary and depends on the field

Answer

Always an even number

Answer

Always divisible by 3

Question 44

Which condition is necessary for a group G to be solvable?

Accepted Answer

Every subgroup of G is solvable

Answer

G is cyclic

Answer

G is abelian

Answer

G has no elements of order greater than 2

Question 45

What is the center of a ring R?

Accepted Answer

The set of all elements that commute with every element in R

Answer

The set of all nilpotent elements in R

Answer

The set of all units in R

Question 46

Which definition describes a simple group?

Accepted Answer

A group with no proper normal subgroups

Answer

An abelian group

Answer

A cyclic group

Answer

A group with a trivial center

Question 47

In a field, what is the multiplicative inverse of a nonzero element?

Accepted Answer

Unique

Answer

Not necessarily unique

Answer

Always equal to -1

Answer

Always equal to 1

Question 48

What do the Sylow theorems provide information about?

Accepted Answer

The existence and structure of subgroups of a finite group

Answer

The solvability of a group

Answer

The number of subgroups of a group

Question 49

Which of the following is an example of a non-commutative ring?

Accepted Answer

The ring of matrices with real entries

Answer

The ring of integers

Answer

The ring of polynomials with rational coefficients

Answer

The ring of continuous functions on the real interval [0, 1]

Question 50

Which of the following rings is NOT a field?

Accepted Answer

A ring with unity and zero divisors.

Answer

A ring with unity and no zero divisors.

Answer

A division ring.

Answer

A ring with identity.

Question 51

A field extension E over F is considered algebraic if:

Accepted Answer

Every element of E is algebraic over F.

Answer

E is a vector space over F.

Answer

F is a subfield of E.

Answer

E is a finite extension of F.

Question 52

What is a characteristic of a perfect field?

Accepted Answer

Every algebraic extension is separable.

Answer

Every finite field extension is Galois.

Answer

Every subgroup of the multiplicative group is cyclic.

Answer

Every irreducible polynomial is separable.

Question 53

Let G be a group and H be a subgroup of G. Which of the following statements is **not** always true?

Accepted Answer

H is a normal subgroup of G.

Answer

H is closed under the group operation of G.

Answer

The inverse of every element in H is also in H.

Answer

The identity element of G is in H.

Question 54

What is the order of the element (1 2 3)(4 5) in the symmetric group S5?

Accepted Answer

6

Answer

10

Answer

3

Answer

5

Question 55

Which of the following functions is a homomorphism from the group of integers under addition (Z, +) to the group of non-zero real numbers under multiplication (R*, ·)?

Accepted Answer

f(n) = 2^n

Answer

f(n) = n + 1

Answer

f(n) = sin(n)

Answer

f(n) = n^2

Question 56

What is the characteristic of the field Z/5Z?

Accepted Answer

5

Answer

∞

Answer

0

Answer

1

Question 57

Let G be a finite group and H be a subgroup of G. What is the relationship between the order of G, the order of H, and the index of H in G?

Accepted Answer

The order of G is equal to the product of the order of H and the index of H in G.

Answer

The order of G is always greater than the order of H.

Answer

The order of G is equal to the sum of the order of H and the index of H in G.

Question 58

Which of the following is a field extension of the field of rational numbers Q?

Accepted Answer

Q(√2)

Answer

R[x]

Answer

Z

Answer

Z[x]

Question 59

Let G be a group and H be a subgroup of G. Which of the following statements is always true?

Accepted Answer

The identity element of G is also the identity element of H.

Answer

The order of H is equal to the order of G.

Answer

H is a normal subgroup of G.

Question 60

Which of the following rings is an integral domain?

Accepted Answer

The ring of integers, Z.

Answer

The zero ring, {0}.

Answer

The ring of 2x2 matrices over the real numbers, M2(R).

Answer

The ring of integers modulo 6, Z6.

Question 61

Consider the ring of polynomials with real coefficients, R[x]. What is the degree of the polynomial f(x) = (x^2 + 1)(x - 2)^3?

Accepted Answer

5

Answer

3

Answer

6

Answer

2

Question 62

Which of the following sets is a field?

Accepted Answer

The set of complex numbers, C.

Answer

The set of even integers, 2Z.

Answer

The ring of 2x2 matrices over the integers, M2(Z).

Answer

The set of integers, Z.

Question 63

What is the characteristic of the field Z_7 (integers modulo 7)?

Accepted Answer

7

Answer

1

Answer

Infinite

Answer

0

Question 64

Let f: G → H be a group homomorphism. What is the kernel of f?

Accepted Answer

{g ∈ G | f(g) = eH}, where eH is the identity element of H.

Answer

The image of f.

Answer

{h ∈ H | f(g) = h for some g ∈ G}.