I came across a problem asserting that the C-tensor product of C and C and the R-tensor product of C and C are isomorphic as Q-modules. How does this begin to make sense as they do not have the same dimension over R? The first is isomorphic to R^{2} and the second is isomorphic to R^{4} over R. Also, I'm struggling with tensors. Does anyone have a good source I should check out? (I've tried many)
Over Q these vector spaces are infinite dimensional. They will be isomorphic if the cardinality of their bases is the same - which I think is the cardinality of the Continuum. The tensor product is a way to extend the field of scalars of a vector space. A real vector space of dimension n (n can be infinite) becomes a complex vector space of dimension n when tensored with C. It becomes a real vector space of twice the dimension over R. If one views R as a 1 dimensional vector space over itself with single basis vector v, the all elements are of the form rV for real numbers,r. Over C the basis is also v and the resulting vector space is 1 dimensional over C. But over R the basis is V and iV and so is 2 dimensional over R.