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