So, I believe that, given a ring spectrum $R$ and an $R$-module $A$, we say that $A$ is "free" if $A\simeq \vee_IR$, i.e. some indexed wedge of copies of $R$. Now, as far as I understand, we can consider a ring spectrum $F$ a "field" if every $F$-module is free (do we need the wedge to be finite?).
I believe that a large class of examples of such fields are the Morava K-theories for a given prime, $K(n)$. Are the Morava $K$-theories the only "fields"? Specifically, do the Morava K-theories contain every Eilenberg-MacLane spectrum $\mathbb{H}F$ for $F$ a field?
If not, is there some other general classification of the fields in brave new algebra?
Many thanks to you!
Edit: One other thought. Is there a correspondence between $K(n)$ at a prime $p$ and $\mathbb{F}_{p^n}$? This is a total guess and completely unsubstantiated, but it would make sense I guess....