Fields in Abstract Algebra: Definition and Examples

Question 1

Which of the following is an example of a finite field with a prime number of elements?

Accepted Answer

The set of integers modulo 7

Answer

The set of rational numbers

Answer

The set of real numbers

Answer

The set of complex numbers

Question 2

In a field, the additive identity element is:

Accepted Answer

Unique and equal to the multiplicative identity

Answer

Unique and not equal to the multiplicative identity

Answer

Not unique and equal to the multiplicative identity

Answer

Not unique and not equal to the multiplicative identity

Question 3

For a subset E of a field F to be a subfield, which condition must be met?

Accepted Answer

E is closed under addition and multiplication, and contains the additive and multiplicative identities of F.

Answer

E is non-empty and closed under addition, but does not necessarily contain the multiplicative identity of F.

Answer

E is non-empty and closed under multiplication, but does not necessarily contain the additive identity of F.

Answer

E is closed under addition only.

Question 4

Which of these sets is NOT a field under standard addition and multiplication?

Accepted Answer

Integers

Answer

Rational numbers

Answer

Real numbers

Answer

Complex numbers

Question 5

Which of the following is NOT an example of a field?

Accepted Answer

Integers

Answer

Rational numbers

Answer

Real numbers

Answer

Complex numbers

Question 6

The set of all rational numbers under addition and multiplication operations is an example of a:

Accepted Answer

Field

Answer

Group

Answer

Ring

Answer

Vector space

Question 7

Division by zero is NOT a closed operation in the field of real numbers because:

Accepted Answer

Division by zero

Answer

Addition

Answer

Multiplication

Answer

Subtraction

Question 8

Which algebraic structure is a generalization of a field?

Accepted Answer

Ring

Answer

Group

Answer

Vector space

Answer

Affine space

Question 9

According to the definition, a field is:

Accepted Answer

a set with two operations, addition and multiplication, that satisfy specific properties

Answer

a ring with unity

Answer

a set of numbers that can be added, subtracted, multiplied, and divided

Answer

a group with unity

Question 10

Which of the following is a subfield of the field of real numbers?

Accepted Answer

the set of rational numbers

Answer

the set of integers

Answer

the set of complex numbers

Answer

the set of polynomials over the rational numbers

Question 11

Which of the following is a characteristic of a finite field?

Accepted Answer

its order is a power of a prime number

Answer

its order is a prime number

Answer

it has no multiplicative identity

Answer

it is not a subfield of any other field

Question 12

Which of the following is NOT a field?

Accepted Answer

the set of polynomials over the integers

Answer

the set of integers modulo 5

Answer

the set of rational numbers

Answer

the set of real numbers

Question 13

Which of the following is the definition of a field in abstract algebra?

Accepted Answer

A commutative ring with unity whose nonzero elements form a multiplicative group

Answer

A non-commutative ring with unity

Answer

A ring without unity whose nonzero elements form a multiplicative group

Question 14

Which of the following is NOT a property of a field?

Accepted Answer

Non-closure under multiplication

Answer

Associativity under addition

Answer

Existence of a multiplicative inverse for every nonzero element

Answer

Commutativity under multiplication

Question 15

Which of the following is an isomorphism between the field of rational numbers and the field of real numbers?

Accepted Answer

x -> x

Answer

x -> 2x

Answer

x -> 1/x

Question 16

Which of the following is a characteristic of a field?

Accepted Answer

The smallest positive integer n such that nx = 0 for all x in the field

Answer

The number of elements in the field

Answer

The set of all generators of the field

Question 17

Which of the following is a finite field?

Accepted Answer

GF(p^n)

Answer

C

Answer

Q

Answer

R

Question 18

Which of the following is an ordered field?

Accepted Answer

A field with a total order that is compatible with the field operations

Answer

A field with a total order that is not compatible with the field operations

Answer

A field with a partial order that is compatible with the field operations

Question 19

Which of the following statements best defines a field in abstract algebra?

Accepted Answer

A mathematical system with addition, multiplication, and inverses, where multiplication distributes over addition, and every nonzero element has a multiplicative inverse.

Answer

A mathematical system with only addition and multiplication, but no inverses.

Answer

A mathematical system with a hierarchy of operations, where multiplication takes precedence over addition.

Question 20

Which of the following is a prime example of a field?

Accepted Answer

The set of rational numbers (Q)

Answer

The set of polynomials (P)

Answer

The set of integers (Z)

Answer

The set of matrices (M)

Question 21

In which area of mathematics are fields commonly applied?

Accepted Answer

Number theory

Answer

Topology

Answer

Calculus

Answer

Geometry

Question 22

Which characteristic is unique to finite fields?

Accepted Answer

The number of elements is always a prime number or a power of a prime number.

Answer

They are subfields of the field of real numbers.

Answer

They are vector spaces over a finite field.

Answer

They are isomorphic to the field of rational numbers.

Question 23

Which action results in a transcendental extension of a field?

Accepted Answer

Adjoining a root of an irreducible polynomial to the field.

Answer

Adjoining a rational number to the field.

Answer

Adjoining an algebraic number to the field.

Question 24

Which is a key attribute of the algebraic closure of a field?

Accepted Answer

It contains all algebraic elements over the field.

Answer

It is always isomorphic to the field of complex numbers.

Answer

It is necessarily the smallest field containing the original field.

Question 25

Identify the set that qualifies as a field:

Accepted Answer

The set of rational numbers

Answer

The set of even integers

Answer

The set of odd integers

Answer

The set of integers

Question 26

Which of the following is a proper subfield of the field of real numbers?

Accepted Answer

The set of rational numbers

Answer

The set of quaternions

Answer

The set of integers

Answer

The set of complex numbers

Question 27

Determine the characteristic of the field of rational numbers:

Accepted Answer

0

Answer

2

Answer

1

Answer

3

Question 28

In which mathematical domain do fields find significant application?

Accepted Answer

Number theory

Answer

Statistics

Answer

Geometry

Answer

Topology

Question 29

Which property is not characteristic of the field of complex numbers?

Accepted Answer

Being an ordered field

Answer

Algebraic closure

Answer

Infinite elements

Answer

Completeness

Question 30

What defines a Galois field?

Accepted Answer

A field with a finite number of elements

Answer

A field that is algebraically closed

Answer

A field with an infinite number of elements

Answer

A field that is complete

Question 31

Which theorem establishes a fundamental relationship between fields and domains?

Accepted Answer

Every field is an integral domain

Answer

Every domain is a field

Answer

Every ring is a field

Answer

Every field is a ring

Question 32

Identify a central concept within the theory of fields:

Accepted Answer

Extension field

Answer

Module

Answer

Ideal

Answer

Subring

Question 33

In abstract algebra, what is the most precise definition of a field?

Accepted Answer

A commutative ring with unity having multiplicative inverses for all non-zero elements

Answer

An algebraic structure endowed with two binary operations which are associative and commutative

Answer

A set with two operations, addition and multiplication, that satisfy specific properties

Question 34

Which of the following sets is a subfield of the field of real numbers?

Accepted Answer

The set of rational numbers (ℚ)

Answer

The set of integers (ℤ)

Answer

The set of complex numbers (ℂ)

Answer

The set of quaternions (ℍ)

Question 35

In abstract algebra, what is a practical application of fields?

Accepted Answer

The construction of finite fields used in coding theory

Answer

Solving differential equations in calculus

Answer

Calculating probabilities in statistics

Question 36

In a field, what is the additive inverse of an element 'a'?

Accepted Answer

-a

Answer

a^2

Answer

-1 * a

Answer

1/a

Question 37

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

Accepted Answer

1/a

Answer

-1 * a

Answer

a^2

Answer

-a

Question 38

Which of the following is a correct statement about the relationship between fields and integral domains?

Accepted Answer

Every field is an integral domain, but not every integral domain is a field

Answer

Integral domains are a subset of fields

Answer

Fields and integral domains are equivalent algebraic structures

Answer

Every integral domain is a field, but not every field is an integral domain

Question 39

What is a distinctive characteristic of a finite field?

Accepted Answer

It has a finite number of elements

Answer

Its elements are all rational numbers

Answer

It has an infinite number of elements

Answer

It is not closed under the four basic arithmetic operations

Question 40

In abstract algebra, what defines a field?

Accepted Answer

A set with two operations, addition (+) and multiplication (*), satisfying specific axioms like associativity, commutativity, distributivity, identity, and inverse.

Answer

A set with two operations, but they don't need to satisfy any specific axioms.

Answer

A set with one operation (+ or *) that fulfills certain axioms.

Question 41

Identify a subfield of the field of real numbers.

Accepted Answer

The set of rational numbers.

Answer

The set of polynomials.

Answer

The set of complex numbers.

Answer

The set of integers.

Question 42

Which statement about fields is true?

Accepted Answer

Every field is an integral domain.

Answer

Every integral domain is a field.

Answer

Every ring is a field.

Answer

Every group is a field.

Question 43

Which of these operations is NOT defined within a field?

Accepted Answer

Exponentiation

Answer

Addition

Answer

Multiplication

Answer

Subtraction

Question 44

What is a direct consequence of every field having a multiplicative inverse?

Accepted Answer

Division is always possible in a field.

Answer

Multiplication is always commutative in a field.

Answer

Every element in a field is its own additive inverse.

Answer

Addition is always associative in a field.

Question 45

Which property of real numbers is NOT essential for it to be considered a field?

Accepted Answer

Order

Answer

Associativity of addition and multiplication

Answer

Commutativity of addition and multiplication

Answer

Distributivity of multiplication over addition

Question 46

What is the defining characteristic of a field in abstract algebra?

Accepted Answer

A set of elements with two operations, addition and multiplication, that satisfy specific axioms.

Answer

A set of elements with one operation, addition, that satisfies specific axioms.

Answer

A set of elements with no specific requirements for operations.

Answer

A set of elements with one operation, multiplication, that satisfies specific axioms.

Question 47

Which of the following is a defining property of a field?

Accepted Answer

Every non-zero element has a multiplicative inverse.

Answer

The multiplicative identity element is the same as the additive identity element.

Answer

The field has a finite number of elements.

Answer

Every element is its own multiplicative inverse.

Question 48

Which statement about the relationship between fields and other algebraic structures is true?

Accepted Answer

A field is a ring with unity.

Answer

A field is a group with unity.

Answer

A group is a field with unity.

Answer

A ring is a field with unity.

Question 49

Which of the following is a finite field?

Accepted Answer

The set of integers modulo 5

Answer

The set of rational numbers

Answer

The set of complex numbers

Answer

The set of real numbers

Question 50

Which of the following is a subfield of the field of real numbers?

Accepted Answer

The set of rational numbers

Answer

The set of complex numbers

Answer

The set of integers

Answer

The set of finite fields