additive and multiplicative groups of a field are not isomorphic
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.
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.