This is from Dummit and Foote Abstract Algebra.

Prove that the additive and multiplicative groups of a field are never isomorphic.

Hint from the book:

Consider three cases: when

is finite, when

and when

in

.

The finite part is not bad.

let

then

but

because 0 does not have an inverse in the multiplicative group.

For the next case I think contradiction is the way to go so I supposed that there exists an isomorphism

and that

I know that

becuase the identity must map to the identity. Also that

and for any

I know that inverses must get mapped to inverses by an isomorphism.

So I don't see how to get a contradiction So I know I am missing something.

Thanks

TES