To prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups. [duplicate]

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• if the additive group contains an element of order $2$ then the multiplicative does not.

• if the additive group contains no element of order $2$ then the multiplicative does.

Quid's answer is in one sense the same as what I was about to post when I read it. But the way I phrased it may make it clear to some people first learning the subject, in a way that quid's might not.

Suppose an isomorphism $\varphi$ from the multiplicative group to the additive group exists. In a field in which $-1\ne1$, we have $(-1)^2=1$ and so $\varphi(-1)+\varphi(-1)=\varphi(1)=0$. This is a field in which $2\ne0$, so it is permissible to divide both sides of the equality $2\varphi(-1)=0$ by $2$ and get $\varphi(-1)=0$. That puts $-1$ in the kernel of the homomorphism $\varphi$, which, being an isomorphism, should have only $1$ in its kernel.

In a field in which $-1=1$ one uses a different argument.