Galois Theory: Master the Concepts with Engaging Quizzes

Question 1

Indicate which condition is essential for a polynomial equation to be solvable by radicals?

Accepted Answer

The Galois group of the equation should be cyclic.

Answer

The degree of the equation should be at most 2.

Answer

The equation should have distinct roots.

Answer

The coefficients of the equation should be rational.

Question 2

Which property is not characteristic of a normal extension?

Accepted Answer

The Galois group of E over F is abelian.

Answer

Every irreducible polynomial in F[x] splits within E[x].

Answer

E is a splitting field for some polynomial in F[x].

Answer

F is the fixed field of some subgroup of Gal(E/F).

Question 3

Which of the following is a consequence of the Fundamental Theorem of Galois Theory?

Accepted Answer

Every Galois group is isomorphic to a subgroup of a permutation group.

Answer

Every finite subgroup of a permutation group is a Galois group.

Answer

Every polynomial equation is solvable by radicals.

Answer

Every field is a Galois extension of its subfield.

Question 4

In Galois theory, a field extension is a Galois extension if it:

Accepted Answer

Is a normal and separable extension.

Answer

Is a finite extension with its Galois group being abelian.

Answer

Is any finite extension.

Answer

Is any normal extension.

Question 5

The Galois group of a polynomial equation is:

Accepted Answer

The group of all automorphisms of the splitting field of the equation.

Answer

The group of all permutations of the roots of the equation.

Answer

The group of all permutations of the coefficients of the equation.

Answer

The group of all polynomials that have the same roots as the equation.

Question 6

The Fundamental Theorem of Galois Theory states that:

Accepted Answer

Every finite Galois extension is a normal and separable extension, and its Galois group is isomorphic to the group of automorphisms of the extension field.

Answer

Every polynomial equation is solvable by radicals.

Answer

The degree of a Galois extension is equal to the order of its Galois group.

Answer

The Galois group of a polynomial equation is always solvable.

Question 7

Calculate the order of the Galois group of the polynomial equation x^4 - 2x^2 + 1 = 0:

Accepted Answer

8

Answer

2

Answer

4

Answer

16

Question 8

The inverse Galois problem asks:

Accepted Answer

Whether every finite group can be realized as the Galois group of a polynomial equation.

Answer

Whether every finite field can be constructed as the splitting field of a polynomial equation.

Answer

Whether every Galois extension is normal and separable.

Answer

Whether every polynomial equation possesses a Galois group.

Question 9

Which of the following properties accurately describes a Galois group?

Accepted Answer

A Galois group is a set of automorphisms of a field extension.

Accepted Answer

A Galois group can aid in determining the solvability of polynomial equations.

Answer

A Galois group is invariably finite in size.

Answer

A Galois group is always an abelian group.

Question 10

Let E be a Galois extension of F and G be a subgroup of the Galois group Gal(E/F). What is the fixed field of G?

Accepted Answer

The smallest subfield of E that contains F and is fixed by every element of G.

Answer

The largest subfield of E that is fixed by every element of G.

Answer

The union of all the subfields of E that are fixed by some element of G.

Answer

The intersection of all the subfields of E that are fixed by some element of G.

Question 11

Which of the following is a consequence of the Fundamental Theorem of Galois Theory?

Accepted Answer

The Galois group of a Galois extension is uniquely determined by the extension.

Answer

Every finite extension of a field is Galois.

Answer

The number of subfields of a Galois extension is equal to the order of the Galois group.

Answer

The base field of a Galois extension is always algebraically closed.

Question 12

Let E be a Galois extension of F and α be an element of E. Which of the following is equivalent to the statement that α is separable over F?

Accepted Answer

The minimal polynomial of α over F has distinct roots.

Answer

The Galois group Gal(E/F) acts transitively on the roots of the minimal polynomial of α over F.

Answer

The extension F(α)/F is Galois.

Answer

The field E is algebraically closed.

Question 13

Let F be a field and E be a finite extension of F. The group of all field automorphisms of E that fix F is denoted by:

Accepted Answer

Gal(F, E)

Answer

Gal(E, F)

Answer

Aut(E, F)

Answer

Hom(E, F)

Question 14

Let F be a field and let E be a finite Galois extension of F with Galois group G. Then, the order of G is equal to:

Accepted Answer

[E:F]

Answer

|E| - |F|

Answer

|F| - |E|

Answer

|Aut(E/F)|

Question 15

Which of the following is a key application of Galois Theory?

Accepted Answer

Determining the solvability of polynomial equations

Answer

Constructing regular polygons

Answer

Classifying finite simple groups

Answer

Solving differential equations

Question 16

Let F be a finite field of order q. Then, the Galois group of the polynomial x^q - x over F is:

Accepted Answer

C_q

Answer

Z_q

Answer

A_q

Answer

S_q

Question 17

Which of the following statements about the Galois group of a polynomial equation is **FALSE**?

Accepted Answer

The Galois group is always abelian.

Answer

The Galois group is a group of automorphisms of the splitting field.

Answer

The Galois group determines the solvability of the polynomial equation by radicals.

Answer

The order of the Galois group is equal to the degree of the splitting field over the base field.

Question 18

A field extension K/F is considered a Galois extension if:

Accepted Answer

K is the splitting field of a polynomial over F, and the Galois group of K/F acts transitively on the roots of that polynomial.

Answer

K is algebraically closed.

Question 19

What is the Galois group of the polynomial x⁴ - 2 over the field of rational numbers (ℚ)?

Accepted Answer

The dihedral group of order 8 (D₄)

Answer

The symmetric group of degree 4 (S₄)

Answer

The cyclic group of order 4 (Z₄)

Answer

The alternating group of degree 4 (A₄)

Question 20

What is the degree of the splitting field of the polynomial x⁴ + 1 over the field of rational numbers (ℚ)?

Accepted Answer

8

Answer

4

Answer

16

Answer

2

Question 21

The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between:

Accepted Answer

Subgroups of the Galois group and intermediate fields of the extension.

Answer

Subfields of the base field and subgroups of the Galois group.

Answer

Elements of the Galois group and roots of the polynomial.

Question 22

Let K be a Galois extension of F with Galois group G. The order of the fixed field of a subgroup H of G is:

Accepted Answer

[K:F]/|H|

Answer

[K:F] * |H|

Answer

[K:F]

Answer

|H|

Question 23

Which of the following statements is TRUE about the Galois group of a separable polynomial?

Accepted Answer

The Galois group acts transitively on the roots of the polynomial.

Answer

The Galois group is isomorphic to the symmetric group on the roots.

Answer

The Galois group is always abelian.

Question 24

For a field extension K/F, what is the necessary and sufficient condition for K/F to be a normal extension?

Accepted Answer

Every irreducible polynomial over F that has a root in K splits completely in K.

Answer

K is algebraically closed.

Answer

K is a finite extension of F.

Question 25

How does Galois theory provide a framework for understanding the solvability of polynomial equations by radicals?

Accepted Answer

It establishes a link between the solvability of a polynomial equation by radicals and the structure of its Galois group, specifically its solvability by subgroups.

Answer

It provides a method to directly find the roots of any polynomial equation.