Let $E/K$ be a finite field extension. Show that there is a finite extension $L/E$ such that $L/K$ is normal.

Question

Let $E/K$ be a finite field extension. Show that there is a finite extension $L/E$ such that $L/K$ is normal.

Is my proof correct?

Let $p\in K[X]$ be irreducible with the set of roots $\{\alpha_1,\dots,\alpha_n\}$ in the algebraic closure $\overline K$. I claim that $L=E(\alpha_1,\dots,\alpha_n)$ satisfies the conditions above:

1. Obviously $L$ is the splitting field of $p$ so $L/K$ is normal.
2. We have that $$[L:E]=[E(\alpha_1,\dots,\alpha_n):E]=\underbrace{[E(\alpha_1,\dots,\alpha_n):E(\alpha_1,\dots,\alpha_{n-1})]}_{\leq\deg(p)}\cdots\underbrace{[E(\alpha_1):E]}_{\leq\deg(p)}\\\leq\deg(p)^n<\infty$$ because the minimal polynomials of $\alpha_i$ on the fields above are divisors of the minimal polynomial $p$ of $\alpha_i$ on $K$.