Remark

— 1) Let L be an extension of K and f a non-constant polynomial in K[X]. By e) of Prop. 3 and V, p. 32, Cor. 2 it comes to the same to suppose that f is separable, whether considered as element of K[X] or of L[X]. On the other hand, it may well happen that f is irreducible in K[X] but not in L[X].

2) Let $f \in K[X]$; we know (IV, p. 13, Prop. 13) that there exist irreducible polynomials $f_1, \ldots, f_m$ in $K[X]$ such that $f = f_1 \ldots f_m$. Let $\Omega$ be an algebraic closure of $K$; since an irreducible polynomial $g \in K[X]$ is the minimal polynomial over $K$ of each of its roots in $\Omega$, two distinct irreducible polynomials in $K[X]$ have no common root in $\Omega$. Condition d) of Prop. 3 now shows that $f$ is separable if and only if the polynomials $f_1, \ldots, f_m$ are separable and pairwise distinct.