Definition 2

— Let K be a commutative field. We say that $f \in K[X]$ is irreducible if $\deg f \geq 1$ and f is not divisible by any polynomial g such that $0 < \deg g < \deg f$. It comes to the same to say that $\deg f \geq 1$ and the only divisors of f in $K[X]$ are the scalars $\neq 0$ and the products of f by scalars $\neq 0$. Since the relation $(f) \subset (g)$ means that g divides f, we see that the irreducible polynomials of $K[X]$ may also be defined as the polynomials f such that the ideal $(f)$ is maximal (I, p. 104).

Let $f, g \in K[X]$. Iff is irreducible, it is clear that either f and g are relatively prime or that f divides g. If f and g are irreducible, either f and g are relatively prime or each is a product of the other by a scalar $\neq 0$. In particular, two distinct irreducible monic polynomials are relatively prime.