TITLE: Tensor product of fields over integers
QUESTION [8 upvotes]: Inspired by this question we ask: Is there a name for each of the following properties about fields? what are some examples other than $\mathbb{Q}$? 

A field $K$ with the property that $K\otimes_{\mathbb{Z}} K$ is a field.
A field $K$ is isomorphic to $K\otimes_{\mathbb{Z}} K$ (as field, or equivalently as rings)

The tensor products are taken over $\mathbb{Z}$ (the tensor product of $\mathbb{Z}$-modules, i.e., Abelian  groups).

REPLY [22 votes]: Here is a self-contained argument. First, as Jeremy Rickard observes, $K \otimes K \cong K \otimes_k K$, where $k$ is the prime subfield of $K$ (so $\mathbb{Q}$ if $K$ has characteristic zero and $\mathbb{F}_p$ if $K$ has characteristic $p$). If $K \otimes_k K$ is a field, then as Denis Nardin observes, the multiplication map
$$K \otimes_k K \xrightarrow{m} K$$
must be injective, and since it's surjective it must be an isomorphism. But $K$ has dimension $1$ as a $K$-vector space, while $K \otimes_k K$ has dimension $\dim_k K$. It follows that $\dim_k K = 1$, hence that $K = k$.