Suppose we have fields $F \subset E \subset K$ and $\alpha \in K$. Suppose also that $K$ is algebraic over $E$. Let $\alpha \in K$. Then we have $p(\alpha)=0$ for some $p(x) = a_{0} + a_{1}x+ \dots+a_{n}x^n \in E[x]$.
The claim is that $|F(a_{0}, a_{1}, \dots a_{n}): F| \leq \displaystyle \prod_{i=0}^{n} |F(a_{i}):F| \lt \infty$.
I am not sure why this inequality holds. Would appreciate if someone has an explanation!