Let $K \subset L \subset M$. Let $f(x)$ be the minimal polynomial for $\alpha$ over $M$. Moreover, suppose $f(x)\in L[x]$, then is the minimal polynomial for $\alpha$ over $L$ also $f(x)$?
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Yes. Let's write $g(x)$ for the minimal polynomial of $\alpha$ over $L$. Since $f(x) \in L[x]$ has $\alpha$ as a root, we have $g \mid f$. But $L[x] \subset M[x]$, so by the same reasoning over $M$ we have $g \mid f$. Since both polynomials are monic, we have equality.