Examples of Fields in Mathematics: Rational, Real, and Complex Numbers

Question 1

Which set does NOT form a field under the given operations?

Accepted Answer

Integers under addition and multiplication

Answer

Rational numbers under addition and multiplication

Answer

Real numbers under addition and multiplication

Answer

Complex numbers under addition and multiplication

Question 2

Which property is NOT a defining characteristic of a field?

Accepted Answer

Division is always defined and results in a unique element

Answer

Addition is associative

Answer

Multiplication is commutative

Answer

Multiplicative inverses exist for non-zero elements

Question 3

In the field of rational numbers, which operation is NOT closed?

Accepted Answer

Division by zero

Answer

Addition

Answer

Subtraction

Answer

Multiplication

Question 4

What is the additive identity in the field of complex numbers?

Accepted Answer

0

Answer

i

Answer

-1

Answer

1

Question 5

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

Question 6

Which field has characteristic 2?

Accepted Answer

The field of integers modulo 2

Answer

The field of rational numbers

Answer

The field of complex numbers

Answer

The field of real numbers

Question 7

What is the degree of the irreducible polynomial (x^2 + 1) over the field of rational numbers?

Accepted Answer

2

Answer

1

Answer

3

Question 8

Which consequence follows from the Fundamental Theorem of Algebra?

Accepted Answer

Every non-constant polynomial with complex coefficients has at least one root.

Answer

Every polynomial with real coefficients has a real root.

Answer

Every polynomial with rational coefficients has a rational root.

Question 9

Identify the key differences between fields, rings, and groups. Specifically, highlight the properties that fields share with both rings and groups.

Accepted Answer

Fields share the associative property with groups and the distributive property with rings.

Answer

Fields differ from rings by having a multiplicative identity but not an additive identity.

Answer

Fields distinguish themselves from groups by lacking a multiplicative inverse.

Answer

Fields are distinct from rings as they do not obey the commutative property for multiplication.

Question 10

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

Accepted Answer

Finite field

Answer

Field of rational numbers

Answer

Field of real numbers

Answer

Field of complex numbers

Question 11

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

Accepted Answer

Rational numbers

Answer

Complex numbers

Answer

Field of p-adic numbers

Answer

Finite field with 8 elements

Question 12

Which of the following is a consequence of the fundamental theorem of algebra?

Accepted Answer

Every polynomial with complex coefficients has a complex root

Answer

Every polynomial has a real root

Answer

Every polynomial with integer coefficients has an integer root

Answer

Every polynomial with rational coefficients has a rational root

Question 13

Which of the following sets forms a field under addition and multiplication?

Accepted Answer

Rational numbers

Answer

Irrational numbers

Answer

Integers

Answer

Natural numbers

Question 14

Which of the following is a nonzero divisor in any field?

Accepted Answer

Any non-zero element

Answer

0

Answer

1

Answer

2

Question 15

What is the multiplicative identity in the field of complex numbers?

Accepted Answer

1

Answer

0

Answer

i

Answer

-1

Question 16

What is the degree of the field extension of the field of rational numbers to the field of complex numbers?

Accepted Answer

2

Answer

Infinite

Answer

0

Answer

1

Question 17

Which of the following is a quotient field?

Accepted Answer

Field of fractions of a polynomial ring over a field

Answer

Field of real numbers

Answer

Field of rational numbers

Answer

Field of complex numbers

Question 18

What is the characteristic of the finite field with 9 elements?

Accepted Answer

3

Answer

0

Answer

2

Answer

1

Question 19

What is a consequence of the associativity property of a field under multiplication?

Accepted Answer

The grouping of factors does not affect the product.

Answer

Multiplication is distributive over addition.

Answer

Multiplication is commutative.

Question 20

Which of the following operations is not closed in the field of rational numbers?

Accepted Answer

Division by zero

Answer

Subtraction

Answer

Multiplication

Answer

Addition

Question 21

What is the multiplicative inverse of 3 in the field of rational numbers?

Accepted Answer

1/3

Answer

3

Answer

-3

Answer

0

Question 22

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

Accepted Answer

5

Answer

0

Answer

-5

Answer

1

Question 23

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

Accepted Answer

It is algebraically closed

Answer

It is a subfield of the real numbers

Answer

It is ordered

Answer

It is finite

Question 24

Which of the following is an application of fields?

Accepted Answer

Cryptography

Answer

Biology

Answer

History

Answer

Physics

Question 25

Which of the following is NOT a property of the rational numbers?

Accepted Answer

Closure under division

Answer

Associativity of multiplication

Answer

Commutativity of addition

Answer

Distributive property

Question 26

What is the multiplicative identity element in the field of complex numbers?

Accepted Answer

1

Answer

i

Answer

0

Answer

-1

Question 27

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

Accepted Answer

The set of integers modulo 7

Answer

The set of all rational numbers

Answer

The set of all real numbers

Answer

The set of all complex numbers

Question 28

Which of the following operations is NOT closed under the field of integers?

Accepted Answer

Division

Answer

Multiplication

Answer

Addition

Answer

Subtraction

Question 29

Which of the following is the multiplicative identity element in the field of rational numbers?

Accepted Answer

1

Answer

i

Answer

0

Answer

-1

Question 30

Which of the following is an application of fields in computer science?

Accepted Answer

Cryptography

Answer

Calculus

Answer

Statistics

Answer

Topology

Question 31

Which of the following statements about the field of complex numbers is true?

Accepted Answer

It is an algebraically closed field.

Answer

It is a finite field.

Answer

It is not a field.

Answer

It is a subfield of the field of real numbers.

Question 32

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

Accepted Answer

The field of integers

Answer

The field of natural numbers

Answer

The field of complex numbers

Answer

The field of real numbers

Question 33

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

Answer

x -> x^2

Question 34

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

Accepted Answer

Its order is a prime power.

Answer

It is not a field.

Answer

It is closed under subtraction.

Answer

It is an infinite field.

Question 35

Which of the following is an application of finite fields in mathematics?

Accepted Answer

Coding theory

Answer

Statistics

Answer

Calculus

Answer

Topology

Question 36

Which of the following sets forms a field under the usual operations of addition and multiplication?

Accepted Answer

Rational numbers

Answer

Irrational numbers

Answer

Natural numbers

Answer

Integers

Question 37

Which of the following operations is not closed under the field of real numbers?

Accepted Answer

Division by zero

Answer

Addition

Answer

Subtraction

Answer

Multiplication

Question 38

Which of the following is a subfield of the complex numbers?

Accepted Answer

Real numbers

Answer

Rational numbers

Answer

Natural numbers

Answer

Integers

Question 39

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

Accepted Answer

Field with prime number of elements

Answer

Transcendental number field

Answer

Real number field

Answer

Algebraic number field

Question 40

What is the symbolic representation of the field of rational numbers?

Accepted Answer

Q

Answer

C

Answer

R

Answer

Z

Question 41

Which of the following is not an axiom of a field?

Accepted Answer

There exists an element without a multiplicative inverse

Answer

There exists a multiplicative identity element

Answer

There exists an additive identity element

Question 42

Which of the following is NOT a field under the standard operations?

Accepted Answer

Integers

Answer

Real numbers

Answer

Rational numbers

Answer

Complex numbers

Question 43

Which operation is not well-defined in any field?

Accepted Answer

Division by zero

Answer

Addition

Answer

Multiplication

Answer

Subtraction

Question 44

Which of the following is a significant application of fields in mathematics?

Accepted Answer

Algebraic geometry

Answer

Numerical analysis

Answer

Descriptive statistics

Answer

Calculus

Question 45

Which of the following statements accurately describes a property of the field of rational numbers?

Accepted Answer

It is a dense field, meaning between any two rational numbers, there exists another rational number.

Answer

It is closed under exponentiation.

Answer

It contains all complex numbers.

Answer

It is a finite field.

Question 46

Which of the following fields has a characteristic of 2, meaning the additive identity is added to itself twice to obtain the additive identity?

Accepted Answer

Binary field

Answer

Field of integers

Answer

Field of complex numbers

Answer

Field of real numbers

Question 47

Which of the following fields is algebraically closed, meaning every non-constant polynomial with coefficients in the field has a root in the field?

Accepted Answer

Complex numbers

Answer

Real numbers

Answer

Rational numbers

Answer

Finite fields

Question 48

Which of the following is a direct consequence of the fact that the field of complex numbers is algebraically closed?

Accepted Answer

Every polynomial with complex coefficients has a complex root.

Answer

Every equation in the field of complex numbers has a solution.

Answer

The complex numbers form a complete ordered field.

Answer

The complex numbers are a finite field.

Question 49

Which of the following is NOT a field?

Accepted Answer

Integers

Answer

Real numbers

Answer

Rational numbers

Answer

Complex numbers

Question 50

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

Accepted Answer

Real numbers

Answer

Rational numbers

Answer

Integers

Answer

Natural numbers

Question 51

Which of the following is an application of fields in mathematics?

Accepted Answer

Algebraic geometry

Answer

Trigonometry

Answer

Calculus

Answer

Probability theory

Question 52

Which of the following statements about the field of real numbers is true?

Accepted Answer

It is an ordered field

Answer

It is a non-Euclidean field

Answer

It is a finite field