Finite Fields in the world of Ring Spectra

[michael...@gmail.com]

So V and I were talking about my favorite algebra theorem: a finite
division algebra is a field. I am curious about how to generalize this
to the world of spectra. During the Galois theory seminar, we tossed
around the idea that division algebra should be modeled in topology by
a condition like "retracts of finite R-modules are finite R-modules".
Commutative is easy: E_{\infty}. My question is simply what we should
use for the "finite" part. If we use finite type in homotopy, then it
fails (the Morava K-theories mess things up). If we use finite in
homotopy, then we get the theorem, but only because it is the algebra
statement. My current guess is finite type (from a homology point of
view), and the statement would look something like:

If D is a finite division ring spectrum, then L_{HZ}(D) is a field
spectrum.

Comments? Other guesses?

M

[Tyler Lawson]

Hi Mike,

The proof I know of that finite division rings are fields is something
to the effect of: Every finite subgroup of a field is cyclic.

It might be that one way to generalize this theorem would be to look
for A_\infty ring spectra R such that \Omega^\infty R, with the
A_\infty multiplicative structure, admits a unique E_\infty structure.

Eilenberg-Maclane spectra for finite fields have such a property. What
other A_\infty structures would be so rigid?