Master Field Operations in Abstract Algebra: Interactive Learning Quizzes

Question 1

In the field of rational numbers, what is the multiplicative inverse of -3?

Accepted Answer

-1/3

Answer

3

Answer

-3

Answer

1/3

Question 2

Which of the following is an example of a set that does NOT form a field?

Accepted Answer

The set of integers with the usual operations

Answer

The set of rational numbers with the usual operations

Answer

The set of real numbers with the usual operations

Answer

The set of complex numbers with the usual operations

Question 3

Which field has the smallest number of elements?

Accepted Answer

The field Z₂

Answer

The field of rational numbers

Answer

The field of real numbers

Answer

The field of complex numbers

Question 4

What is the order of the field Z₃?

Accepted Answer

3

Answer

2

Answer

4

Answer

6

Question 5

In a field, which of the following properties holds true regarding addition and multiplication?

Accepted Answer

Both addition and multiplication are associative.

Answer

Only addition is associative.

Answer

Only multiplication is associative.

Answer

Neither operation is associative.

Question 6

Consider the field of real numbers. What is the additive inverse of the element 5?

Accepted Answer

-5

Answer

5

Answer

0

Answer

1

Question 7

In the field of complex numbers, what is the multiplicative inverse of the complex number (1 + i)?

Accepted Answer

(1 - i)/(1 + i)

Answer

(1 - i)/2

Answer

(1 + i)/2

Answer

(1 + i)/(1 - i)

Question 8

Which property of fields ensures that subtraction can always be performed?

Accepted Answer

Existence of additive inverses

Answer

Existence of multiplicative inverses

Answer

Associativity of addition

Answer

Distributivity of multiplication over addition

Question 9

Consider the following set of operations on rational numbers:

Accepted Answer

Addition, subtraction, multiplication, division

Answer

Addition, subtraction, multiplication

Answer

Addition, subtraction, division

Answer

Multiplication, division

Question 10

Which of the following is an example of a non-field?

Accepted Answer

The set of all positive integers

Answer

The set of all real numbers

Answer

The set of all rational numbers

Answer

The set of all complex numbers

Question 11

Which of the following statements about fields is true?

Accepted Answer

Every field has a unique additive identity element.

Answer

Every field has multiple multiplicative identity elements.

Answer

Fields can have non-unique additive inverse elements.

Answer

Fields do not have multiplicative inverses for all non-zero elements.

Question 12

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

Accepted Answer

It contains a finite number of elements.

Answer

It forms a group under addition and multiplication operations.

Answer

It has zero divisors.

Answer

It is isomorphic to the real field.

Question 13

Which expression represents the multiplicative identity element in a field?

Accepted Answer

a * 1

Answer

a * 0

Answer

a * a^(-1)

Answer

a + a

Question 14

Which of the following is not a binary operation, as required for a field?

Accepted Answer

Absolute value

Answer

Addition

Answer

Subtraction

Answer

Multiplication

Question 15

Which of the following is not a property of field multiplication?

Accepted Answer

Distributive over division

Answer

Associative

Answer

Commutative

Answer

Identity element

Question 16

Consider the field of rational numbers Q. Which of the following is the multiplicative identity?

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 17

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

Accepted Answer

Contains a finite number of elements

Answer

Has an infinite number of elements

Answer

Is not commutative

Answer

Does not have an identity element

Question 18

Consider the polynomial ring Z[x]. Is it a field under the usual operations of addition and multiplication?

Accepted Answer

No, because it does not have a multiplicative inverse for every element

Answer

Yes, because it satisfies all field axioms

Answer

Yes, but only if the coefficients are restricted to be integers

Answer

No, because it is not commutative under multiplication

Question 19

Which of the following fields has the smallest number of elements?

Accepted Answer

Z/2

Answer

Z/3

Answer

Z/4

Answer

Z/5

Question 20

Which of the following is not a subfield of the field R of real numbers?

Accepted Answer

Z/2

Answer

Q

Answer

C

Answer

Z

Question 21

Identify which of the following sets is NOT a field:

Accepted Answer

Z/n (integers modulo n with n = 10)

Answer

Q (rational numbers)

Answer

R (real numbers)

Answer

C (complex numbers)

Question 22

Determine if the following expression aligns with a field axiom: (a * b) * c = a * (b * c).

Accepted Answer

True

Answer

False

Question 23

The set of integers with standard addition and multiplication does NOT form a field. Which property is missing?

Accepted Answer

Existence of multiplicative inverses

Answer

Closure under addition

Answer

Associativity of multiplication

Answer

Commutativity of addition

Question 24

Do the even integers, using standard addition and multiplication, form a field?

Accepted Answer

No, multiplicative inverses don't exist for all even integers.

Answer

Yes, they are closed under addition and multiplication, and have identities for both.

Question 25

Which property is not applicable to addition in a field?

Accepted Answer

Addition is not associative.

Answer

An additive identity element exists.

Answer

Addition is commutative.

Answer

Every element has an additive inverse.

Question 26

Which statement accurately describes the distributive property in a field?

Accepted Answer

Multiplication has precedence over addition.

Answer

Multiplication is only distributive over addition for positive numbers.

Answer

Addition has precedence over multiplication.

Question 27

In the field Z/6, calculate the result of 4 + 5.

Accepted Answer

3

Answer

9

Answer

4

Answer

5

Question 28

What is a defining characteristic of a finite field?

Accepted Answer

It contains a finite number of elements.

Answer

It lacks a multiplicative identity element.

Answer

It is always non-commutative.

Answer

It is primarily used in algebra.

Question 29

In a field, which element plays the role of the multiplicative identity?

Accepted Answer

1

Answer

-1

Answer

0

Answer

Any element except zero

Question 30

Which operation is not closed in the field of rational numbers?

Accepted Answer

Division by zero

Answer

Addition

Answer

Multiplication

Answer

Subtraction

Question 31

In the field of real numbers, what is the correct order of operations in the expression (3 + 4) * 5?

Accepted Answer

Parentheses first, then multiplication

Answer

Addition first, then multiplication

Answer

Parentheses and multiplication have equal precedence

Answer

Multiplication first, then parentheses

Question 32

In the field of real numbers, the additive inverse of -7 is:

Accepted Answer

7

Answer

0

Answer

-14

Answer

14

Question 33

The multiplicative identity in the field of rational numbers is:

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 34

Which of the following equations demonstrates the associative property of addition in a field?

Accepted Answer

(a + b) + c = a + (b + c)

Answer

a + b = b + a

Answer

a - b = b - a

Answer

a(b + c) = ab + ac

Question 35

In the field of complex numbers, the product of (3 + 4i) and (2 - 5i) is:

Accepted Answer

22 - 11i

Answer

11 - 22i

Answer

-5 + 3i

Answer

5 - 3i

Question 36

Which of the following is a consequence of the distributive property in fields?

Accepted Answer

a(b + c) = ab + ac

Answer

a - b = b - a

Answer

(a + b) + c = a + (b + c)

Answer

a + b = b + a

Question 37

The quotient of 12 divided by -3 in the field of integers is:

Accepted Answer

-4

Answer

4

Answer

15

Answer

-12

Question 38

Which of the following sets is NOT a field?

Accepted Answer

The set of integers

Answer

The set of real numbers

Answer

The set of complex numbers

Answer

The set of rational numbers

Question 39

In the field of real numbers, the inverse of 1/2 is:

Accepted Answer

2

Answer

1/4

Answer

0

Answer

-1/2

Question 40

Which of the following operations are well-defined in a field?

Accepted Answer

Addition, subtraction, multiplication, and division

Answer

Exponentiation and logarithm

Answer

Sine and cosine

Answer

Subtraction and division only

Question 41

In a field F, which of the following statements is NOT always true for elements a, b, and c in F?

Accepted Answer

If a * b = a * c, then b = c.

Answer

a + (b + c) = (a + b) + c.

Answer

a * (b + c) = (a * b) + (a * c).

Answer

There exists an element 0 in F such that a + 0 = a.

Question 42

What is the additive inverse of 5 in the field of real numbers?

Accepted Answer

-5

Answer

0

Answer

5

Answer

1/5

Question 43

Given a field F and non-zero elements a and b in F, which statement is always true?

Accepted Answer

The product a * b is also non-zero.

Answer

The difference a - b is always non-zero.

Answer

The sum a + b is also non-zero.

Answer

The quotient a / b is always defined.

Question 44

Which of the following sets, with standard addition and multiplication, forms a field?

Accepted Answer

The set of rational numbers, Q

Answer

The set of even integers

Answer

The set of integers, Z

Answer

The set of natural numbers, N

Question 45

In a field, what is the result of multiplying any element 'a' by the additive identity element, denoted as 0?

Accepted Answer

0

Answer

a

Answer

1

Question 46

In a field, which property ensures that for any elements a and b, a * b = b * a?

Accepted Answer

Commutativity of multiplication

Answer

Existence of multiplicative inverses

Answer

Associativity of multiplication

Answer

Distributivity of multiplication over addition

Question 47

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

Accepted Answer

Multiplication is not always commutative.

Answer

Addition and multiplication are associative.

Answer

Addition and multiplication are distributive.

Answer

Every non-zero element has a multiplicative inverse.

Question 48

Determine the additive identity element in the field of complex numbers.

Accepted Answer

0

Answer

1

Answer

-1

Answer

i

Question 49

Which property of field operations guarantees that the sum of any two elements in a field is also an element of that field?

Accepted Answer

Closure under addition

Answer

Distributivity of multiplication over addition

Answer

Associativity of addition

Answer

Commutativity of addition

Question 50

Find the additive inverse of 3 in the field Z/5Z.

Accepted Answer

2

Answer

-3

Answer

1

Answer

3

Question 51

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

Accepted Answer

Commutativity

Answer

Associativity

Answer

Closure

Answer

Distributivity over addition

Question 52

In the field of rational numbers, what is the multiplicative inverse of -2/3?

Accepted Answer

-3/2

Answer

1/2

Answer

2/3

Answer

3/2

Question 53

Which of the following statements about fields is true?

Accepted Answer

Every nonzero element has a multiplicative inverse.

Answer

Fields are closed under subtraction.

Answer

Every element has an additive inverse.

Question 54

In the field Z/4Z, find the value of 3 + 2.

Accepted Answer

1

Answer

2

Answer

3

Answer

4

Question 55

Which of the following is an example of a finite field?

Accepted Answer

Z/5Z

Answer

C

Answer

R

Answer

Q

Question 56

In the field of real numbers, what is the distributive property of multiplication over addition?

Accepted Answer

a(b + c) = ab + ac

Answer

a + (b + c) = (a + b) + c

Answer

a(bc) = (ab)c

Answer

a + b = b + a

Question 57

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

Accepted Answer

It has a prime number of elements.

Answer

It is closed under subtraction.

Answer

Every element has a square root.

Answer

It is an infinite field.