Tensor Product of C with itself over R.

[Monobrow]

I am trying to prove that C\otimesC (taken over R) is equal to C^2. The method I have seen is to show the following equivalences:
C\otimesC = C\otimes(R[T]/<T^2+1>) = C[T]/<T^2+1> = C.
(All tensor products taken over R).

The only part I am having trouble with is showing that C\otimes(R[T]/<T^2+1>) = C[T]/<T^2+1>. I have tried to show this by using the universal property of the tensor product. First I defined a bilinear map from CXR[T]/<T^2+1> to C[T]/<T^2+1> by sending an element (z,f(T)) to zf(t) (where I am omitting the cosets). This then induces a linear map from the tensor product. However, I cannot seem to find an inverse for this map.

Any help would be greatly appreciated.

 

[fresh_42]

The general method is called complexification and works for any real vector space.
https://en.wikipedia.org/wiki/ComplexificationOr more detailed and better written:
https://kconrad.math.uconn.edu/blurbs/linmultialg/complexification.pdf