Field Theory – About the Additive Group and the Multiplicative Group of a Field

Question

Let $F$ be a field. When happens that the additive group of $F$ is isomorphic to the multiplicative group?

It is easily to work out that $F$ must have characteristic $0$, but then what?

Answer 1

If you already have that char$\,\Bbb F=0\;$ then what does $\;-1\;$ map to? This is an element of order two in $\;\Bbb F^*\;$ so it must map to an element of order two in $\;\Bbb F\;$ .