In Nilpotence in Stable Homotopy Theory II, Hopkins–Smith define a field spectrum to be a ring spectrum $E$ such that $E_*X$ is a free $E_*$-module for all spectra $X$. They then show that any field spectrum $E$ has the homotopy type of a wedge of suspensions of a Morava K-theory $K(p,n)$ for some $n\in\mathbb{N}$ and some prime $p$. In this sense, the Morava K-theories are the prime fields in stable homotopy theory.

This is very different from the first definition of fields we usually see in commutative algebra, that $k$ is a field if $k^\times=k\setminus\{0\}$, and so I was wondering if one could adapt this definition to work also for ring spectra. These days I saw an answer which suggested to me a way to do precisely this, but it requires a bit of background.

One of the possible approaches to the field with one element $\mathbb{F}_{1}$ is to start by defining $\mathbb{F}_{1}$-modules as pointed sets, observing that the category of pointed sets admits a symmetric closed monoidal structure $(\mathsf{Sets}_*,\wedge,S^0)$, and then defining $\mathbb{F}_{1}$-algebras as the monoids in that category. Those turn out to be “monoids with zero”―pairs $(A,0_{A})$ with $A$ a monoid and $0_{A}$ an element of A such that, for each $a\in A$, we have \begin{align*} 0_{A}\cdot a &= 0_{A},\\ a\cdot 0_{A} &= 0_{A} \end{align*} for all $a\in A$. For example, every semiring or ring $(R,\cdot,+,1_{R},0_{R})$ has an underlying $\mathbb{F}_{1}$-algebra, the pair $((R,\cdot_{R},1_{R}),0_{R})$.

The field with one element $\mathbb{F}_{1}$ is then defined as the initial $\mathbb{F}_{1}$-algebra. Concretely, it consists of the set $\{0,1\}$ equipped with the monoid with zero structure where $0_{\mathbb{F}_{1}}=0$, $1_{\mathbb{F}_{1}}=1$, and with multiplication table given by

A more homotopical way of thinking about this is that, just as $(\mathbb{N},+,0)$ is the free $\mathbb{E}_\infty$-monoid on one element in the category $\mathsf{Sets}$, the underlying $\mathbb{F}_{1}$-module (i.e. pointed set) of the field with one element $\mathbb{F}_{1}$, namely $(\{0,1\},1)$, is the free $\mathbb{E}_0$-monoid on one element in $\mathsf{Sets}$. Moreover, when we consider $\mathbb{E}_{\infty}$-monoids on these categories, $\mathbb{F}_{1}$ becomes the unit for the tensor product of $\mathbb{F}_{1}$-algebras, while $\mathbb{N}$ and $\mathbb{Z}$ become the unit for the tensor products of semiring and rings.

Now, while the category $(\mathsf{Sets}_*,\wedge,S^0)$ is not Cartesian monoidal, it turns out that, by a result of of Péroux–Shipley, any comonoid in it is of the form $X^+$ for $X$ a set. So in effect we can consider group objects in $(\mathsf{Sets}_*,\wedge,S^0)$, as for a monoid $(A,0_{A})$ in it to have a Hopf monoid structure is in fact a property, rather than extra structure, and it moreover corresponds to the condition $A^\times=A\setminus\{0_{A}\}$.

This leads to the following characterisation of fields among all rings:

• A ring $k$ is a field iff its underlying $\mathbb{F}_{1}$-algebra is a Hopf $\mathbb{F}_{1}$-algebra.

Now, we may run this same procedure for ring spectra.

Firstly, we may define a spectral $\mathbb{E}_{k}$-algebra over $\mathbb{F}_{1}$ to be an $\mathbb{E}_{k}$-monoid in $\mathcal{S}_*$, with a similar definition of spectral homotopy associative/commutative $\mathbb{F}_{1}$-algebras. For example, every semi/ring spectrum $E$ has an underlying spectral $\mathbb{E}_k$-algebra over $\mathbb{F}_{1}$, the space $\Omega^\infty E$.

Once again, the $\infty$-category $(\mathcal{S}_*,\wedge,S^0)$ is not Cartesian monoidal, but being an $\mathbb{E}_{k}$-comonoid in $\mathcal{S}_*$ is nevertheless a property, rather than extra structure. So we can consider the following definition, together with its analogues for homotopy associative/commutative ring spectra:

Definition. An $\mathbb{E}_{k}$-field spectrum is an $\mathbb{E}_{k}$-ring spectrum $E$ whose underlying spectral $\mathbb{E}_{k}$-algebra over $\mathbb{F}_{1}$, $\Omega^{\infty}E$, is an $\mathbb{E}_{k}$-Hopf algebra in $\mathcal{S}_*$.

Questions:

• How does this definition compare to the one of Hopkins–Smith?
• What are some examples of these objects?
• Do we have once again have prime fields when using this definition, and, if so, are the Morava K-theories among them?