degree of f
f ∈ K[X] as f =SUM^n_i=0 aiXi with ai ∈ K.
If an =! 0 then the degree of f is n, deg(f) = n
Division Theorem for Polynomials
Let K be a field and take f, g ∈ K[X] with g =! 0. Then there are (unique) q, r ∈ K[X] with
f = qg + r and deg(r) < deg(g) or r = 0
(q is the quotient and r the remainder when f is divided by g.)
An element a ∈ K is a root (or zero) of f ∈ K[X] if f(a) = 0.
greatest common divisor
The greatest common divisor (or highest common factor) 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.
Write d = gcd(f, g).
This polynomial is defined only up to a non-zero scalar multiple so, if we want a unique gcd then we can insist that d be monic (ie. coefficient of highest power
of X is equal to 1).
An element r ∈ R is irreducible if r is not invertible and
if, whenever r = st either s or t is invertible
Elements r, s ∈ R are associated if s = ur for some invertible element u ∈ R.
Unique Factorisation Domain (UFD)
A commutative domain R is said to be a Unique Factorisation Domain (UFD), if every non-zero, non-invertible element of R has a unique factorisation as a product of irreducible elements.
’Uniqueness’ here means up to rearrangement of factors and associated factors
Principal Ideal Domain (PID)
a commutative domain in which every ideal is principal (that is, is generated by a single element).