What's the tensor product of the real line with itself, as abelian groups?

This is just R. R is a field extension of Q (a transcendental one), so naturally we can view R as a Q-vector space. Thus, R is isomorphic to Q\^I where I is some uncountably infinite indexing set. It follows that R tensor R is isomorphic to (Q\^I) tensor (Q\^I), and as products distribute over tensors, this is isomorphic to I\^2 copies of Q tensor Q. Q tensor Q (over Z) is just Q, so we have Q\^{I\^2}. I\^2 is still uncountable, so the cardinality of I\^2 is the same as the cardinality of I. Thus,

R tensor R = (Q\^I) tensor (Q\^I) = (Q tensor Q)\^{I\^2} = (Q)\^I = R

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