Galois Theory Quiz

A field extension E/F is called a Galois extension if:

• It is finite and normal

• It is separable and normal

• It is only normal

• It is only separable

The degree of the extension [E:F][E : F] is defined as:

• The number of automorphisms

• The dimension of E as a vector space over F

• The number of roots in E

• The number of minimal polynomials

A polynomial is said to be separable over a field if:

• It has only real roots

• It has distinct roots in its splitting field

• It has repeated roots

• It is irreducible

The splitting field of a polynomial is:

• The field where it is irreducible

• The smallest field extension over which it splits completely

• The field with complex numbers

• The base field

The Galois group of a polynomial over a field F is:

• A group of permutations of roots

• A group of field automorphisms of the splitting field fixing F

• A group of matrices

• A group of polynomials

The identity automorphism maps:

• All elements to their roots

• Each element to itself

• Zero to one

• Elements to their negatives

An extension E/F is normal if:

• Every irreducible polynomial over F that has a root in E splits completely in E

• E is a subfield of F

• F is algebraically closed

• It has transcendental elements

The order of the Galois group equals the:

• Degree of the minimal polynomial

• Number of real roots

• Degree of the extension

• Number of elements in the base field

Every Galois extension is:

• Algebraic only

• Normal and separable

• Transcendental

• Infinite

A field automorphism of E fixing F is:

• An isomorphism from E to F

• A bijection from E to itself leaving elements of F unchanged

• A zero function

• A constant map