Here, $\sigma : K \to \bar{L}$ is a ring homomorphism and $\bar{L}$ the algebraic closure of $L$. $\mu_{\alpha}$ is the minimal polynomial of $\alpha$.
$\sigma(\mu_{\alpha})$ is the polynomial that's created from $\mu_{\alpha}$ by applying $\sigma$ to its coefficients.
Since $\alpha$ is separable, $\mu_{\alpha}$ factors into pairwise different factors in $\bar{L}$ by definition.
Why does then $\sigma(\mu_{\alpha})$ also factor into pairwise different linear factors in $\bar{L}$? Is that because $\sigma$ is injective?