Field spectra and Eilenberg–MacLane spectra

Question

Apparently thanks to a theorem of Hopkins and Smith, every field spectrum splits into a wedge of Morava K-theories, where we allow the cases $K(0) = H \mathbb{Q}$ and $K(\infty) = H \mathbb{F}_p$. I do not understand this result. What about the Eilenberg-MacLane space $H\mathbb{R}$? Either it is not a field spectrum (which weirds me out) or it is a wedge of Morava K-theories (which also weirds me out). How does this work?

Answer 1

$H\mathbb{R}$ is a wedge of Morava K-theories: it is a wedge of uncountably many copies of $K(0)=H\mathbb{Q}$. This is because as an abelian group, $\mathbb{R}$ is just a direct sum of uncountably many copies of $\mathbb{Q}$ (pick a basis for $\mathbb{R}$ as a vector space over $\mathbb{Q}$).

More generally, in the case of Eilenberg-MacLane spectra, this theorem is just the elementary fact that every field is a vector space over $\mathbb{Q}$ or $\mathbb{F}_p$ for some prime $p$, and thus as an abelian group is a direct sum of copies of them.