Corollary 1

— Let $K$ be a field, $L$ a separable extension of $K$ and $f$ a polynomial in $K[X]$. If $f$ has no multiple factors in $K[X]$, it has also no multiple factors in $L[X]$.

If $f$ has no multiple factor in $K[X]$, the quotient ring $K[X]/(f)$ is reduced; for $f$ is a product of irreducible polynomials $f_i$ relatively prime in pairs, hence

K[X]/(f) is isomorphic, by I, p. 110, Prop. 9, to the product of fields K[X]/(f_i). By Prop. 5 the ring L[X]/(f), being isomorphic to L ⊗_K K[X]/(f ), is reduced; if g is a non-constant polynomial of L[X] such that g^2 divides f, then the residue class of fg^{-1} in L[X]/(f) is a non-zero nilpotent element; hence f has no multiple factors in L[X].