Linear transformations of fields

Question

Let $K$ be field. Show if $f: K \to K$ is linear transformation, then there exists $a \in K$ such that for every $x \in K$, $f(x) = ax$

I don't know how to prove it but for instance $a = 1$ satisfy the conditions (I think so, not sure)

Answer 1

Hints: it must be that

$$\;\forall\,x\in K\;,\;\;f(x)=f(x\cdot1)=xf(1)$$

So what do you think $\;a\;$ would (must) be?