I read somewhere (and would like to ask for confirmation) saying that simple extensions of an algebraic element are always of the form $K[X]/(m)$ where $m$ is the minimal polynomial (which is the only irreducible polynomial?) containing that element. This appears to me to be exactly the same (up to isomorphism) as the splitting field of its minimal polynomial. Is it true that there is a natural identification between them two?
Also, may I conclude hence, that for any irreducible polynomial $m$ in $K[X]$ and its roots $\alpha_1, ..., \alpha_n\in\overline{K}$, we have $K(\alpha_1)=(\cong) ...= K(\alpha_n)\cong K[X]/(m)$? In particular (I saw similar claims, not sure why), in the case of a separable extension, $\{\alpha_1, ..., \alpha_n\}$ as well as $\{\alpha_i^1, ..., \alpha_i^n\}, \forall i$ always form bases of the extension?