In section 3.2.1 of these notes it is said the splitting field of a polynomial $f$ over $\Bbbk$ is a field $L=\Bbbk(\alpha_1,\dots ,\alpha_n)$ over which $f$ splits ($(\alpha_1,\dots ,\alpha_n)$ are the roots of $f$).
I don't understand a detail in one direction of the proof: why does $f$ split over $\Bbbk(\alpha_1,\dots ,\alpha_n)$? I mean why can we write a polynomial as a product of linear factors if its roots are all present in the field (ring?)?