# Tensor product of fields

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

## 3 thoughts on “Tensor product of fields”

1. HINT: consider the multiplication map $K \otimes_{\mathbb{Z}} K \to K$.

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2. 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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3. nachodarago says:

There is a closely related formula by Grothendieck:
Let $K|k, L|k$ be field extensions. Then $\dim_{\mathrm{Krull}}(K\otimes_k L)=\min(\mathrm{trdeg}_k(K),\mathrm{trdeg}_k(L))$.

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