5) Polynomial Rings and Factorisation Flashcards
What is the Division Theorem for Polynomials
What is a root of a field
An element a ∈ K is a root of f ∈ K[X] if f(a) = 0
Under what condition is an element of a field a root
Let K be a field, let f ∈ K[X] and let a ∈ K. Then a is a root of f iff X − a is a factor of f
Describe the proof that an element of a field is a root iff X − a is a factor of f
What is the greatest common divisor of a polynomial
The greatest common divisor of polynomials f, g is a polynomial d such that d divides f and g and, if h is any polynomial dividing both f and g, then h divides d
What is a monic greatest common divisor
A unqiue gcd such that gcd(f, g) = 1
What is the Division Algorithm for Polynomials
What is the ideal generated by two polynomials equivalent to
What do we know about the ideal of every polynomial ring
Every ideal of a polynomial ring K[X] is principal
Describe the proof that every ideal of a polynomial ring is principal
What are irreducible and associated elements
How can you prove a polynomial is irreducible using reduction modulo n
- If the polynomial were reducible, then reducing it modulo every n (except specific exceptions) would result in a reducible polynomial
- Find a value of n such that the polynomial remains irreducible when reduced modulo n.
- If you can find such an n, the original polynomial must be irreducible
What is a Unique Factorisation Domain (UFD)
A commutative domain R is said to be a UFD, if every non-zero, non-invertible element of R has a unique factorisation as a product of irreducible elements.
’Unique’ here means up to rearrangement of factors and associated factors
What is a Principal Ideal Domain (PID)
A commutative domain in which every ideal is principal
Give an example of a Principal Ideal Domain (PID)
The ring Z is a PID since every ideal has the form < n > for some integer n
