The additive and multiplicative groups of a field are never isomorphic

Let $F$ be a field. Prove that as groups, $(F,+)$ and $(F^\times, \cdot)$ are not isomorphic.

First suppose $F$ is finite. Then $|F| = n$ and $|F^\times| = n-1$ for some natural number $n \geq 1$. In particular, no bijection $\Phi : F \rightarrow F^\times$ exists, so that these groups cannot be isomorphic.

Suppose now that $F$ is infinite and that $\Phi : F \rightarrow F^\times$ is a field isomorphism. Note that $\Phi(0) = 1$.

Suppose $1 = -1$. Now $\Phi(1) = a$ for some $a \in F$, and $\Phi(-1) = a^{-1}$. But also $\Phi(-1) = \Phi(1) = a$. So $a^2 = \Phi(1)\Phi(-1) = \Phi(0) = 1$. Thus $a^2 = 1$, and we have $a = \pm 1 = 1$. But then $\Phi(0) = \Phi(1)$, and $\Phi$ is not injective, a contradiction.

Suppose instead that $1 \neq -1$. Now $\Phi(a) = -1$ for some $a \in F$, so that $\Phi(2a) = \Phi(a)^2 = 1$. Since $\Phi$ is injective, $2a = 0$. Thus $a = 0$, and we have $1 = \Phi(0) = -1$, a contradiction.

Thus $(F,+)$ and $(F^\times, \cdot)$ are not isomorphic.