Let f ∈ R[x] be monic and f = ∏i f i as product of irreducibles. Then the leading coeffiicents of the f i are
units, without loss of generality one can multiply with scalars to have that all the f i are monic as well.
Let Q = R/I and θ∶ R[x] → Q[x] be the reduction hod I homomorphism. Then f θ and f i θ are monic as well,
with deg( f i ) = deg( f i θ) and
f θ = ∏ f i θ.
i
However f i θ might not be irreducible, but might factor further. Because of uniqueness of factorization in
polynomial rings, we can obtain the factorization of f θ from ∏i f i θ by factoring the f i θ further. Thus the
degrees of the irreducible factors of f θ can be combined in ways to obtain the degrees of the f i , i.e. the
irreducible factors of f .
This means that the partition of deg f = deg f θ given by the irreducible factors of f θ is a refinement of the
partition given by the irreducible factors of f .