Algebra Quiz

What is a field in mathematics?

• A collection of random symbols

• A type of polynomial

• A commutative ring with unity where every nonzero element is a unit

• A set of numbers with no operations

What does the theorem 'Finite Integral Domains Are Fields' state?

• All finite sets are fields

• Finite integral domains are not fields

• Only infinite sets can be fields

• Finite integral domains are fields

What is a polynomial ring?

• A set of numbers

• A ring of formal symbols over a commutative ring

• A type of field

• A collection of integers

What is an extension field?

• A field that contains another field

• A field with no elements

• A field with only one element

• A type of polynomial

What does it mean for a polynomial to split in an extension field?

• It has no roots

• It has roots in the extension field

• It is a constant polynomial

• It is irreducible

What is the derivative of a polynomial?

• A polynomial with all coefficients equal to zero

• A polynomial that represents the slope

• A polynomial that has no roots

• A polynomial with no terms

What is the criterion for multiple zeros of a polynomial?

• It is a perfect field

• It is a constant polynomial

• It has a common factor of positive degree

• It has no common factors

What does it mean for a field to be perfect?

• Every polynomial has a root

• It has no elements

• It contains only integers

• It has only one element

What is a splitting field?

• A field with no elements

• A field where a polynomial can be factored completely

• A field that contains only rational numbers

• A field with only one polynomial

What happens if a polynomial has a multiple zero?

• It is a perfect field

• It is always irreducible

• It has a common factor with its derivative

• It has no roots