An exercise about field extension

Question

Let $K/F$ be a field extension, $L$ an intermediate field, $\alpha \in K$ is algebraic over $F$ with minimal polynomial $p(x) \in F[x] $. If $p(x)$ is irreducible in $L[x]$, then $F(\alpha)\cap L=F$. Could you please help me this exercise? Thank you in advance.

Answer 1

Take an element $\beta \in F(\alpha) \cap L$. Then you have that by the Tower Law

$$\deg p(x) = [F(\alpha) : F] = [F(\alpha) : F(\beta)][F(\beta) : F]$$

They key thing here is that $\beta \in L$ so $F(\beta) \subseteq L$. $p$ is irreducible in $L$ so it must be irreducible in $F(\beta)$ and $[F(\alpha) : F(\beta)] = \deg p(x)$.

From the above equality, $[F(\beta) : F] = 1$ so $\beta \in F$. This shows $F(\alpha) \cap L \subseteq F$. But clearly it contains $F$ so equality follows.