Let $(\mathbb K,+,\cdot)$ be a field.
Is there a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot) $ ?
The answer should clearly be negative.
I tried to proceed via contradiction, but this has not led me very far.
Since it's some kind of "trick problem", I'm merely looking for hints.
There is never such an isomorphism. For finite fields this is straightforward because the additive and multiplicative groups have different cardinalities. For infinite fields we can proceed as follows:
If the field $k$ does not have characteristic $2$, then $-1$ is an element of order $2$ in $k^{\times}$. But by hypothesis, since $k$ does not have characteristic $2$, the additive group of $k$ is a vector space over the prime subfield of $k$, which is not $\mathbb{F}_2$, so it has no elements of order $2$. Hence it cannot be isomorphic to the multiplicative group. If $k$ has characteristic $2$, then the additive group of $k$ contains only elements of order $2$. On the other hand, $k$ is infinite, but the equation $x^2 = 1$ admits at most two solutions in a field, so again the additive group cannot be isomorphic to the multiplicative group.