prove $[E(a):E] \le [F(a):F]$

Question

Let $F<E<K$ be field extensions, such that $a \in K$, and $[K:F]<\infty$,

Is it true that $[E(a):E] \le [F(a):F]$?

How can I show this?

Answer 1

Since $[K:F]$ is finite, $a$ is a root of a polynomial in $F[X]$, and this polynomial belongs to $E[X]$, too.