$\mathbb{C}\otimes_\mathbb{C} \mathbb{C} \cong \mathbb{R}\otimes _\mathbb{R} \mathbb{C}$ I am trying to show that $\mathbb{C}\otimes_\mathbb{C} \mathbb{C} \cong \mathbb{R}\otimes _\mathbb{R} \mathbb{C}$ as abelian groups.
I've tried to come up with various maps but gotten nowhere. 
Thanks for any help

The groups $\mathbb C\otimes_{\mathbb C}\mathbb C$ and $\mathbb R\otimes_{\mathbb R}\mathbb C$ are both isomorphic to $\mathbb C$. More general, for every $\mathbb C$-vector space $V$ you have $\mathbb C\otimes_{\mathbb C}V\cong V\cong\mathbb R\otimes_{\mathbb R}V$ because for any unitary (commutative) ring $A$ and any $A$-module $M$ you have $A\otimes_AM\cong M$.
A proof of this isomorphism.
The function $A\times M\to M:(a,x)\mapsto ax$ is $A$ bilinear, hence it induces a group homomorphism $h:A\otimes_A M\to M:a\otimes x\to ax$. Since $1\in A$ this is surjective, while injectivity follows from $0=h(\sum_ia_i\otimes x_i)=\sum_i a_ix_i$ implies $\sum_ia_i\otimes x_i=1\otimes(\sum_ia_i x_i)=0$.
