Show that $n$ is a divisor of $[L:K]$

Question

Let $L/K$ be a field extension.

I want to show that if the extension $L/K$ is finite and $a\in L$ has a minimal polynomial of degree $n$, then $n$ is a divisor of $[L:K]$.

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I have done the following:

We have that $[K(a):K]=\deg \text{Irr}(a,K)=n$ and $K\leq K(a)\leq L$, or not?

Since $L/K$ is finite, we have that $$[L:K]=[K(a):K][L:K(a)]\Rightarrow [K(a):K]\mid [L:K] \Rightarrow n\mid [L:K]$$

Is everything correct? Could I improve something?