Can you construct a field of characteristic $\neq 0, 2$ such that every one of its subrings is also a field? A friend asked me this a few days ago, and I was thinking that it may be impossible, but now I'm not so sure. He suggested a "nonprincipal ultrapower" $(\mathbb{Z}/(2))^{N}$ such that every subring is isomorphic to $\mathbb{Z}/(2)$, but I'm not entirely sure what this means. Any thoughts?

Finite fields fit the bill, since any element is the inverse of one of its powers.   The same goes for algebraic closure of finite fields (since these are unions of finite fields).
Note that if the transcendence degree of your field $k$ over its prime field $\mathbb{F}_p$ is non-zero, then the result does not hold, since if $t$ is transcendent over $\mathbb{F}_p$, then $\mathbb{F}_p[t]$ is a subring of $k$ which is not a field.  This implies that the only examples are finite fields and their algebraic extensions.
