Master Field Extensions in Abstract Algebra: Enhance Your Mathematical Skills

Question 1

Which of the following best defines a field extension?

Accepted Answer

A field that contains another field as a subfield

Answer

A field that is a subset of another field

Answer

A field that is isomorphic to another field

Answer

A field that has a different characteristic from another field

Question 2

What is meant by the Galois group of a field extension?

Accepted Answer

The group consisting of all automorphisms of the field extension

Answer

The group consisting of all isomorphisms between the field extension and its base field

Answer

The group resulting from all permutations of the elements in the field extension

Answer

The group formed by all subgroups of the field extension

Question 3

What is the order of a field extension's Galois group?

Accepted Answer

The degree of the field extension

Answer

The number of elements in the field extension

Answer

The number of automorphisms of the field extension

Answer

The number of subgroups of the field extension

Question 4

Let L/K be a field extension. Which of the following statements is equivalent to L being a separable extension?

Accepted Answer

The minimal polynomial of every element in L over K has no repeated roots

Answer

The characteristic of K is zero

Answer

L is a normal extension

Answer

L is a radical extension

Question 5

Let E/F be a field extension and α ∈ E. Which of the following is the fixed field of α under Gal(E/F)?

Accepted Answer

The smallest subfield of E containing F and α

Answer

The largest subfield of E containing F and α

Answer

Isomorphic to F(α)

Answer

Always a normal extension

Question 6

Which of the following is a fundamental theorem in Galois theory?

Accepted Answer

Both of the above

Answer

Every finite normal extension is Galois

Answer

The Galois group of a Galois extension is isomorphic to the automorphism group of its fixed field

Question 7

Let F be a field extension of K. Which of the following is always true?

Accepted Answer

K is a subset of F

Answer

K is not a subset of F

Answer

F is a subset of K

Answer

K ∩ F = {0}

Question 8

What is the Galois group of the field extension ℝ(x) over ℝ, where x is transcendental?

Accepted Answer

{1}

Answer

ℝ

Answer

ℝ*

Answer

ℤ

Question 9

How can Galois theory be used to determine the solvability of a polynomial equation with coefficients in a given field F by radicals over F?

Accepted Answer

By examining the Galois group of the equation, which provides insights into its solvability.

Answer

By factoring the equation into irreducible polynomials, which can reveal information about its solvability.

Answer

By using the discriminant of the equation, which provides information about the nature of its roots.

Answer

By applying the quadratic formula, which is applicable to quadratic equations and cannot be used for higher-degree equations.

Question 10

Which of the following is a Galois field?

Accepted Answer

The finite field with 2^n elements (?_{2^n})

Answer

The rational numbers (ℚ)

Answer

The real numbers (ℝ)

Answer

The complex numbers (ℂ)

Question 11

Let K be a Galois extension of F. Which of the following is true for the Galois group G(K/F)?

Accepted Answer

It is a finite group.

Answer

It is an infinite group.

Answer

It is always abelian.

Answer

It is isomorphic to the multiplicative group of the field K.

Question 12

Let K be a field extension of F. Which of the following is a necessary and sufficient condition for K to be a normal extension of F?

Accepted Answer

Every irreducible polynomial in F[x] that has a root in K splits completely in K[x].

Answer

Every irreducible polynomial in K[x] that has a root in F splits completely in F[x].

Answer

Every irreducible polynomial in F[x] that has a root in K is irreducible in K[x].

Answer

Every irreducible polynomial in K[x] that has a root in F is irreducible in F[x].

Question 13

Which of the following is an example of a transcendental field extension?

Accepted Answer

ℝ(π)

Answer

ℚ(√2)

Answer

ℂ(ζ3)

Answer

?_{2^n}

Question 14

In field theory, what is the definition of algebraic closure, and why is it important in the study of field extensions?

Accepted Answer

Algebraic closure refers to a field that contains all roots of all polynomials with coefficients in the field.

Answer

Algebraic closure is only applicable to finite fields.

Answer

Multiple distinct algebraic closures can exist for a given field.

Answer

The concept of algebraic closure has no practical relevance in field theory.

Question 15

Provide a precise definition for a field extension.

Accepted Answer

An algebraic structure that extends an existing field, satisfying additional properties

Answer

A mapping between two fields that preserves their operations

Answer

A product of two fields that forms a new field

Answer

A quotient of a polynomial ring over a field that results in a new field

Question 16

Which consequence directly follows from the Primitive Element Theorem?

Accepted Answer

Every field possesses a primitive element.

Answer

Every finite field extension is cyclic.

Answer

All field extensions are finite.

Answer

Irreducible polynomials always have roots in field extensions.

Question 17

Which characteristic uniquely identifies a separable field extension?

Accepted Answer

Its minimal polynomial over the base field is free of perfect squares.

Answer

Its Galois group is commutative.

Answer

Its degree equals the order of its Galois group.

Answer

It is purely transcendental.

Question 18

Consider the field extension ℝ ⊂ ℝ(√2). Which statement accurately describes this extension?

Accepted Answer

The Galois group of ℝ(√2) / ℝ is isomorphic to ℤ₂.

Answer

ℝ(√2) is a normal extension of ℝ.

Answer

ℝ(√2) is purely inseparable over ℝ.

Answer

ℝ(√2) is a finite field extension of ℝ.

Question 19

Which tool is instrumental in studying the arithmetic properties of number fields?

Accepted Answer

Class field theory

Answer

Category theory

Answer

Riemann zeta function

Answer

Modular forms

Question 20

Which of the following is the Galois group of the polynomial x^4 + 1 over the field of rational numbers?

Accepted Answer

The Klein four-group V₄

Answer

The cyclic group of order 4 ℤ₄

Answer

The dihedral group of order 8 D₈

Answer

The symmetric group on 4 letters S₄

Question 21

In a Galois extension E over F, which of the following statements is true?

Accepted Answer

The fixed field of the Galois group is F.

Answer

The fixed field of the Galois group is E.

Answer

The fixed field of the Galois group is a subfield of F.

Answer

The fixed field of the Galois group is a subfield of E.

Question 22

Let F be a field extension of K. Which of the following is true about the relationship between F and K?

Accepted Answer

K is a subspace of F under field operations

Answer

F is a subspace of K under field operations

Answer

F and K have the same order

Answer

F and K have the same characteristic

Question 23

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

Accepted Answer

ℤ/5ℤ

Answer

ℤ

Answer

ℝ

Answer

ℚ

Question 24

What is the Galois group of the field extension ℚ(ζ)/ℚ, where ζ is a primitive 5th root of unity?

Accepted Answer

Cyclic group of order 5

Answer

Cyclic group of order 4

Answer

Symmetric group of order 5

Answer

Dihedral group of order 10

Question 25

What is a practical application of Galois theory?

Accepted Answer

Determining the solvability of polynomial equations by radicals

Answer

Solving systems of linear equations

Answer

Simplifying complex algebraic expressions

Answer

Finding prime numbers

Question 26

Which of the following is an example of a transcendental field extension?

Accepted Answer

Extension of the field of rational functions over a finite field by an exponential function

Answer

Extension of the field of rational numbers by the square root of 2

Answer

Extension of the field of real numbers by the imaginary unit i

Answer

Extension of the field of complex numbers by a primitive nth root of unity

Question 27

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

Accepted Answer

ℚ(√2)

Answer

ℤ (set of integers)

Answer

ℂ (set of complex numbers)

Answer

ℚ² (set of ordered pairs of rational numbers)

Question 28

Let f(x) be an irreducible polynomial over a field F and α be a root of f(x) in a field extension E of F. What can be concluded about [F(α) : F]?

Accepted Answer

It is equal to the degree of f(x)

Answer

α is a transcendental element over F

Answer

F(α) is a normal extension of F

Answer

[F(α) : F] is always equal to 2

Question 29

Which of the following is a Galois field?

Accepted Answer

GF(p^n) (finite field with p^n elements, where p is prime)

Answer

ℤ[x] (ring of polynomials with integer coefficients)

Answer

ℝ (field of real numbers)

Answer

ℚ (field of rational numbers)

Question 30

Which of the following is a fundamental theorem of Galois theory?

Accepted Answer

The Galois group of a finite Galois extension is isomorphic to the group of automorphisms of the extension that fix the base field

Answer

Every finite field is a Galois field

Question 31

Let E be a normal extension of F. Which statement is equivalent to saying that E is Galois over F?

Accepted Answer

The Galois group Gal(E/F) is a transitive group on the set of subfields of E containing F

Answer

E is a separable extension of F

Answer

Gal(E/F) is a solvable group

Question 32

Which statement about insolvability by radicals is true?

Accepted Answer

The general quintic equation cannot be solved by radicals

Answer

All cubic equations can be solved by radicals

Answer

All quartic equations can be solved by radicals

Question 33

Which of the following is NOT a field extension?

Accepted Answer

The set of complex numbers with the restricted addition operation (a + b = a - b)

Answer

The set of all algebraic numbers

Answer

The set of real numbers extended by the imaginary unit i

Answer

The set of rational numbers extended by the square root of 2

Question 34

The Galois group of a field extension K over F is defined as:

Accepted Answer

The group of all automorphisms of K that fix F element-wise

Answer

The group of all isomorphisms from K to F

Answer

The group of all endomorphisms of K that fix F element-wise

Answer

The group of all permutations of K that fix F element-wise

Question 35

The minimal polynomial of an element α in a field extension K over F is:

Accepted Answer

The monic polynomial of least degree in F[x] that has α as a root

Answer

The polynomial of highest degree in F[x] that has α as a root

Answer

The polynomial that has α as its only root

Answer

The polynomial that has α as a multiple root

Question 36

Which of the following statements about the degree of a field extension is true?

Accepted Answer

The degree of a finite field extension is equal to the order of the Galois group

Answer

The degree of a field extension is equal to the number of elements in the extension field

Answer

The degree of a field extension is equal to the number of subfields in the extension field

Answer

The degree of a field extension is always a prime number

Question 37

Which of the following is NOT a field extension of the field of rational numbers, Q?

Accepted Answer

Q[x]/(x^2 + 1)

Answer

Q(√2)

Answer

Q(√3)

Answer

Q(√5)

Question 38

What is the degree of the field extension Q(√2, √3) over Q?

Accepted Answer

4

Answer

3

Answer

1

Answer

2

Question 39

Which of the following is a basis for the field extension Q(√2) over Q?

Accepted Answer

{1, √2}

Answer

{1, √2, √4}

Answer

{1, 2}

Answer

{√2, √4}

Question 40

Consider the polynomial f(x) = x^3 - 2 over Q. What is the splitting field of f(x) over Q?

Accepted Answer

Q(∛2, ω)

Answer

Q(ω)

Answer

Q(√2)

Answer

Q(∛2)

Question 41

Which of the following statements is TRUE about the Galois group of the field extension Q(√2) over Q?

Accepted Answer

It is cyclic of order 2.

Answer

It is trivial.

Answer

It is isomorphic to the dihedral group of order 4.

Answer

It is isomorphic to the symmetric group S3.

Question 42

What is the minimal polynomial of √2 + √3 over Q?

Accepted Answer

x^4 - 10x^2 + 1

Answer

x^4 - 2

Answer

x^3 - 5x

Answer

x^2 - 5

Question 43

If E is a finite extension of F, what is the relationship between the degree of the extension [E:F] and the number of automorphisms of E over F?

Accepted Answer

[E:F] is equal to the order of the Galois group of E over F.

Answer

[E:F] is less than the order of the Galois group.

Answer

[E:F] is greater than the order of the Galois group.

Question 44

The Fundamental Theorem of Galois Theory establishes a correspondence between:

Accepted Answer

Subgroups of the Galois group and subfields of the extension field.

Answer

Polynomials over the base field and subfields of the extension field.

Answer

Elements of the Galois group and elements of the extension field.

Question 45

Given a field extension K of a field F, which statement is always true?

Accepted Answer

F is a subfield of K.

Answer

K is finite if and only if F is finite.

Answer

K and F have the same number of elements.

Answer

K is a subfield of F.

Question 46

How can you construct a field extension K of a field F?

Accepted Answer

Adjoin a root of an irreducible polynomial over F to F.

Answer

Add a random element from a larger field to F.

Answer

Take the union of F and K.

Answer

Take the intersection of F and K.

Question 47

If K is a field extension of F and G is its Galois group, what is the order of G?

Accepted Answer

The order of G is equal to the degree of the extension K over F.

Answer

The order of G is equal to the number of elements in K.

Answer

The order of G is equal to the degree of the minimal polynomial of an element of K over F.

Answer

The order of G is always a prime number.

Question 48

What is a famous unsolved problem in Galois theory?

Accepted Answer

The Inverse Galois Problem.

Answer

Fermat's Last Theorem.

Answer

The Goldbach Conjecture.

Answer

The Riemann Hypothesis.

Question 49

Given a field extension E of a field F, which statement is true?

Accepted Answer

E contains F as a subfield.

Answer

E and F are disjoint sets.

Answer

F contains E as a subfield.

Answer

E is a proper subset of F.

Question 50

What is a necessary condition for a field extension E of F to be a normal extension?

Accepted Answer

Every irreducible polynomial in F remains irreducible in E.

Answer

E is a Galois extension of F.

Answer

E is a finite extension of F.

Answer

E contains all roots of every polynomial in F[x].

Question 51

What is a key consequence of Galois theory?

Accepted Answer

The Galois group of a field extension helps determine the number and types of intermediate fields between the base field and the extension field.

Answer

Every field extension is a normal extension.

Answer

All field extensions are solvable.

Question 52

Given a field F and an irreducible polynomial p(x) in F[x], which of the following is a field extension of F?

Accepted Answer

F[x]/(p(x)).

Answer

F(x).

Answer

Z[x].

Answer

R[x].

Question 53

What is the degree of a field extension?

Accepted Answer

The number of elements in the extension field that are algebraic over the base field.

Answer

The number of elements in the base field.

Answer

The degree of the minimal polynomial of a generating element.

Question 54

What is the Fundamental Theorem of Galois Theory?

Accepted Answer

It establishes a one-to-one correspondence between subgroups of the Galois group and subfields of the field extension.

Answer

It proves the insolvability of the quintic equation.

Question 55

Let F be a field and let G be a finite group. Which of the following is a necessary and sufficient condition for G to be the Galois group of a field extension of F?

Accepted Answer

G is isomorphic to a subgroup of the automorphism group of the field extension.

Answer

G is an abelian group.

Answer

G is a cyclic group.

Question 56

Which of the following is an application of field extensions in cryptography?

Accepted Answer

RSA encryption.

Answer

AES encryption.

Answer

Hashing.

Answer

DES encryption.