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Jan 14, 2019 at 22:30 comment added Wojowu For group structure, it's enough to note the tensor product is a $\mathbb Q$-vector space of size, hence dimension, continuum. No need to consider the same for $\mathbb R$.
Jan 14, 2019 at 21:52 comment added R. van Dobben de Bruyn This only gives the additive group structure. It seems to me that the OP wants to understand it as a ring.
Jan 14, 2019 at 21:35 history answered George C. Modoi CC BY-SA 4.0