Finite field extensions and minimal polynomial

Question

I want to show the following statement:

Let L/K be a finite field extension with $[L:K]=p$ for a prime $p$

Show that $[L:K]$ is simple

Proof:

1) Choose $\alpha\in L$ with $\alpha \notin K$. Then $K(\alpha)$ is stricty greater than $K$.

With the formula $[L:K]=[L:K(\alpha)]\cdot[K(\alpha):K]$ it follows that $[L:K(\alpha)]=p$,because p is a prime.

The assertion follows.

Is my proof correct? (Seems a bit too easy)

Answer 1

You're close. It doesn't follow that $[L:K(\alpha)]=p$, because then $[K(\alpha):K]=1$, hence $K(\alpha)=K$, which contradicts the fact that $K(\alpha)$ is strictly greater than $K$ by your choice of $\alpha$.

Instead, the formula shows $[L:K(\alpha)]$ divides $[L:K]=p$, and since it is not $p$, $[L:K(\alpha)]=1$ since $p$ is prime, so $L=K(\alpha)$, hence $L$ is simple as it's generated by adjoining a single element to $K$.