Proposition. some propositions [1704]

    1. $\alpha \in \overline{F}$ is a multiple root of $f(x) \in F[x]$ if and only if it is a root of both $f(x)$ and $f'(x)$.
    2. $f(x) \in F[x]$ is separable if and only if $\gcd \left( f(x), f'(x) \right) = 1$.
    3. An irreducible polynomial in $F[x]$ is separable if and only if $f'(x) \neq 0$.
    4. If $F$ is a field of characteristic $0$, then every irreducible polynomial in $F[x]$ is separable and every field extension of a field of characteristic $0$ is separable.

    Proof. sketch [1704A]

      1. Omitted.

      2. Let $d(x) = \gcd(f(x), f'(x))$.

        Claim: $\alpha \in \overline{F}$ is a common root of $f(x)$ and $f'(x)$ if and only if $\alpha$ is a root of $d(x)$. Then, just follow part 1.

        Let’s prove the claim.

        $\impliedby$: It is clear that $d(\alpha) = 0 \implies f(\alpha) = f'(\alpha) = 0$.

        $\implies$: Since $F[x]$ is a PID, there exists $a(x), b(x) \in F[x]$ such that $a(x)f(x) + b(x)f'(x) = d(x)$. Then $d(\alpha) = a(\alpha)f(\alpha) + b(\alpha)f'(\alpha) = 0$.

      3. A direct proposition of part 2.

      4. By part 3.

      5. $\operatorname{ch}(F) = 0 \implies f'(x) \neq 0$, then use part 3. For every $K / F$ and $\alpha \in K$, by part 4, $m_{\alpha, F}(x)$ is separable, so $\alpha$ is separable over $F$, hence $K / F$ is separable.