I’m reading a proof of the following fact: if $L/K$ is a finite field extension and $\alpha \in L$, then $\chi=f^{[L:K(\alpha)]}$ where $\chi$ and $f$ are the characteristic and minimal polynomials of $\alpha$, respectively.
The proof goes as follows: we know that if $\{\alpha_1, \ldots, \alpha_n\}$ is a $K$-base of $K(\alpha)$ and $\{\beta_1, \ldots, \beta_m\}$ is a $K(\alpha)$-base of $L$, then the elements $\alpha_i \beta_i$ are a $K$-base of $L$ (so far so good). With this choice of bases, if $M \in M_n(K)$ is the matrix associated to the endomorphism $\varphi_\alpha \in End_K(K(\alpha))$ such that $\varphi_\alpha(\beta)=\alpha \beta$ (multiplication by $\alpha$ in $K(\alpha)$, then the matrix $N\in M_{nm}(K)$ corresponding to the multiplication by $\alpha$ in $L$ is a block-diagonal matrix, each of the blocks corresponding to $M$.
I can see how to conclude from here, but I don’t see explicitely why this last statement holds. Could someone please give some more detail on the precise construction?