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? 


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. 
