Extensions of fields and subfields generated by a field and a set

Question

K ⊆ L ⊆ M be two field extensions and α ∈ M algebraic over K then it is algebraic over L.

Help me prove this please

Answer 1

Since $\alpha \in M$ is algebraic over $K$, it is a zero of some polynomial

$k(x) \in K[x]; \tag 1$

since

$K \subset L, \tag 2$

we have

$K[x] \subset L[x], \tag 3$

which shows that

$k(x) \in L[x]; \tag 4$

but this by definition implies $\alpha$ is algebraic over $K$.