Sanity Check for Simple Extension Proof

[Mandelbroth]

This was an exercise out of Garling's A Course in Galois Theory.

Suppose $L:K$ is a field extension. If $[L:K]$ is prime, then $L:K$ is simple.

I've developed a habit of checking my work for these exercises religiously (the subject matter is gorgeously elegant, so I want to do it justice). I found a much more complicated answer than mine when I googled it, which makes me uncomfortable.

Here's my proof:

Suppose $\alpha\in L$ is not in $K$. Because $[L:K]=[L:K(\alpha)][K(\alpha):K]$ and $[K(\alpha):K]\neq 1$, we must have $[K(\alpha):K]=[L:K]$. Because any two vector spaces with the same dimension over the same field are isomorphic, this completes the proof. []

Is this right?

 

[R136a1]

This should be in the homework forum by the way.

And yes, the proofs seems correct to me.

 

[Mandelbroth]

This should be in the homework forum by the way.

Oops! Indeed, it should. My apologies.

Thank you.