I recently realized for the first time (while teaching my undergraduate abstract algebra course) that simple groups and fields share the analogous property that neither has (nontrivial) factor groups/rings. Likewise, any homomorphism from a simple group or a field is either the 0 map or an embedding (which of course is just a corollary to my previous statement).

1) Has this relationship ever served as motivation for some line of research?

It makes sense that $\mathbb{Z}_p$ is both a simple group and a field.

2) Are there groups analogous to larger fields, say, $\mathbb{Q}$ or $\mathbb{Z}_p[\sqrt{q}]$?

3) On the other hand, there are nonabelian simple groups and, of course, the sporadic groups. Are there fields (or types of fields) then corresponding to the various types of simple groups?

Or maybe this is all just a coincidence and/or dead-end?