Quiz on Polynomials and Galois Theory

A polynomial is solvable by radicals if its roots can be expressed using:

• Matrices

• Rational functions

• Radicals and field operations

• Trigonometric identities

Solvability by radicals is related to the structure of:

• The splitting field

• The group of integers

• The symmetric group

• The Galois group

A key result is that a polynomial is solvable by radicals if and only if:

• Its degree is less than 5

• Its Galois group is solvable

• It has integer coefficients

• It is irreducible

A field extension obtained by adjoining radicals is called a:

• Rational extension

• Radical extension

• Transcendental extension

• Galois extension

A normal extension is an extension in which:

• Every element is algebraic

• Every irreducible polynomial splits completely

• Galois group is trivial

• Extension is purely transcendental

The radical tower of field extensions involves:

• Adjoining rational numbers only

• Adjoining square roots of elements

• Adjoining radicals in successive steps

• Subtracting polynomials

The Galois group of a radical extension is:

• Trivial

• Solvable

• Non-cyclic

• Non-abelian

The quintic equation is not solvable by radicals because:

• It has irrational coefficients

• Its Galois group is not solvable

• It has complex roots

• It is not a polynomial

A solvable group is one that has:

• No subgroups

• A composition series with abelian factors

• Infinite order

• Only normal subgroups

A polynomial of degree 4 is:

• Always solvable by radicals

• Never solvable

• Irreducible over R

• Not a Galois polynomial