Extension Fields and Transcendental Numbers

What is an extension field?

• A field containing a subfield

• A set of numbers closed under addition and multiplication

• A field that is a subset of another field

• A field with more elements than its subfield

If F⊆E , what is the degree of the extension [E:F][E : F]?

• The number of elements in E

• The number of elements in F

• The dimension of E as a vector space over F

• The degree of the minimal polynomial of E

Which of the following is true for a simple extension field?

• It is generated by a single element

• It contains infinitely many elements

• It is always finite

• It is always algebraic

If E is a finite extension of F, then E is:

• Algebraic over FF

• Transcendental over FF

• Both algebraic and transcendental

• Neither algebraic nor transcendental

Which of the following is a property of a Galois extension?

• It is always finite

• It is always algebraic

• It is always normal and separable

• It is always transcendental

Which of the following is true for a finite extension field?

• It is always algebraic

• It is always transcendental

• It is both algebraic and transcendental

• It is neither algebraic nor transcendental

If E is a finite extension of F, then E is:

• Algebraic over F

• Transcendental over F

• Both algebraic and transcendental

• Neither algebraic nor transcendental

Which of the following is a property of a Galois extension?

• It is always finite

• It is always algebraic

• It is always normal and separable

• It is always transcendental

What does it mean for a number to be transcendental over Q?

• It satisfies a polynomial equation with rational coefficients

• It is not real

• It does not satisfy any polynomial equation with rational coefficients

• It is imaginary

Who first proved that ee is transcendental?

• Galois

• Lindemann

• Euler

• Hermite