Show that $[L:K]=1 \Leftrightarrow L=K$ 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 $a=1$ ? 
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Could you ive me a hint for the other direction? 

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.
