Show that $[L:K]=1 \Leftrightarrow L=K$

Question

Let $L/K$ be a field extension.

I want to show that $$[L:K]=1 \Leftrightarrow L=K$$

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I have done the following:

For the direction $\Rightarrow \ : $

Since $[L:K]=1=\text{dim}_KL$ we have that there exist $a\in L$ with $\langle a\rangle$ a $K$-basis of $L$.

So, let $\ell\in L$, then $$\ell=ak, k\in K$$

To get the desired result, can we just take $a=1$ ?

$$$$

Could you ive me a hint for the other direction?

Answer 1

To get the desired result, can we just take $a = 1$?

Yes, you are using that in a one-dimensional vector space, any non-zero vector gives a basis.

Can you give me a hint for the other direction?

You have to show that $K$ is one-dimensional as a vector space over itself.

Answer 2

Show first arrow:
$\Rightarrow)$ If $L/K $ has dimension $1$ we know that $K/(p(x)) $ has dimension $1$ as $K$- vector space so necessarily the degree of $p(x)$ must be $1$ (I'm considering the homomorphism of valutation: $\psi : K[x]\to L $ such that $\psi (p)= p(\alpha )\space \forall \space p \in K[x]$ with $ \alpha \in L $).
Therefore $K/(p(x))=K$ and being $K/(p(x))\cong L$, $K\cong L$.
(We can say are equal).

$\Leftarrow)$ Obvious.