Inspired by this [question](https://mathoverflow.net/questions/82083/when-is-the-tensor-product-of-two-fields-a-field) we ask: Is there a name for each of the following properties about fields? what are some examples other than $\mathbb{Q}$? 


1. A field $K$ with the property that $K\otimes\_{\mathbb{Z}} K$ is a field.
2. 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).



Tensor product of fields over integers
I already wrote this in the comments but I think this might be worth of an answer. I think we can classify all fields $K$ such that $K\otimes K$ is a field.


**Claim** If $K$ is a field such that $K\otimes\_\mathbb{Z}K$ is a field then the multiplication map $K\otimes\_\mathbb{Z} K\to K$ is an isomorphism


In fact the multiplication map is always a surjection because $x\otimes 1$ gets sent to $x$. Moreover all maps of fields are injections, so the map is a bijection. But a bijective map of rings is an isomorphism.


Now the rings $R$ such that the multiplication map $R\otimes\_\mathbb{Z}R\to R$ is an isomorphim have been classified by Bousfield and Kan in their paper [*The core of a ring*](http://www.sciencedirect.com/science/article/pii/0022404972900230). For our purposes we need lemmas 3.6 and 3.7 in said paper.


Now there are two possibilities for our field $K$: either it is of characteristic $p$ or it is of characteristic 0. If it is of characteristic 0 it cannot have any torsion subgroup. Hence from lemma 3.7 it needs to be a localization of $\mathbb{Z}$. Since it is a field it must be $\mathbb{Q}$.


If the characteristic of $K$ is $p$ it coincides with its own torsion subgroup. Hence from lemma 3.6 it is of the form $\mathbb{Z}/n$ for some $n$. Since $K$ is a field it must be $n=p$.


