It is known that the only "fields" in stable homotopy theory, after localizing at a prime $p$, are Eilenberg-Mac Lane spectra for fields and the Morava K-theories (this is true in a few senses: these are the only spectra for which a Kunneth theorem holds on their cohomology theories, they are the only spectra over which every module is a free module, and these are all the minimal classes in the Bousfield lattice). Are other cases known in which all the fields are classified? For instance, if we localize at an arbitrary $E_\infty$-ring spectrum $R$, is there a way of determining which spectra satisfy any of the three criteria mentioned above? Must they be localizations of the "global fields" (by global of course I still mean $p$-local)?

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    $\begingroup$ I suspect the answer is uninteresting for harmonic spectra. $\endgroup$ – Jonathan Beardsley Nov 6 '13 at 3:56
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    $\begingroup$ Your first sentence isn't quite true: what is the case is that any field is a module over some $K(n)$ (including $K(0)=H\mathbb{Q}$ and $K(\infty)=H\mathbb{F}_p$). For instance, for $p>2$, the mod $p$ $K$-theory spectrum is not equal to $K(1)$ but is a module over $K(1)$. $\endgroup$ – Eric Wofsey Nov 6 '13 at 4:02
  • $\begingroup$ I think the answer to this question is essentially "no". There's some recent work of Gepner-Antieau where you can take a flat extension of the sphere spectrum (there aren't many of these- mostly they look like "take a flat extension of the integers and tensor up to S") and then the fields you get are, no surprise, the Morava K-theories and the residue fields of pi_0 of your flat extension. $\endgroup$ – Dylan Wilson Nov 6 '13 at 4:51
  • $\begingroup$ @EricWofsey but every module over $K(n)$ is a wedge of $K(n)$'s yes? $\endgroup$ – Jonathan Beardsley Nov 6 '13 at 14:34
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    $\begingroup$ @DylanWilson why restrict oneself to flat extensions? I'm saying for arbitrary $R$, look at the $R$-local category, and ask for spectra which satisfy any of those criteria. There are obviously a few cases where this is known (localize at $K(n)$, or $E_n$, or $BP$). $\endgroup$ – Jonathan Beardsley Nov 6 '13 at 14:38

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