Existence of a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot)$

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.

Solution 1:

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.