Factorization in Polynomial Rings

We begin with some basic definitions. Definition 1.1. Let f, g ∈ F [x]. We say that f divides g, written f ∣∣g, if there exists an h ∈ F [x] such that g = fh, i.e. g is a multiple of f . Thus, for example, every f ∈ F [x] divides the zero polynomial 0, but g is divisible by 0 ⇐⇒ g = 0. By definition, f is a unit ⇐⇒ f ∣∣1. Recall also that the group of units (F [x])∗ of the ring F [x] is F ∗, the group of units in the field F , and hence the group of nonzero elements of F under multiplication. Thus f divides every g ∈ F [x] ⇐⇒ f divides 1 ⇐⇒ f ∈ F ∗ is a nonzero constant polynomial. Finally note that, if c ∈ F ∗ is a unit, then f ∣∣g ⇐⇒ cf ∣∣g ⇐⇒ f ∣∣cg.