Tensor product of fields

Classify the fields K such that the ring K \otimes_{\mathbb{Z}} K is again a field.

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Tensor product of fields

3 thoughts on “Tensor product of fields

  1. I claim this is true if and only if K is a prime field, that is, K is either the rationals or a finite field of order p, p a prime. Indeed it is easy to see that K\otimes_\Bbb Z K is isomorphic to K\otimes_F K where F is the prime subfield of K. If this is a field, then multiplication (as in Luis’ comment) is injective, from where \dim_F K = (\dim_F K)^2 and \dim_F K=1; whence F=K.

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