= Morava K-theory
{c}
{title2=$K(n)_*$}
At a <prime number> $p$ and height $n\ge1$, <Morava K-theory> is a periodic <generalized homology theory> represented by a <topological spectrum> $K(n)$ with coefficient <graded ring>
$$
K(n)_*=\mathbb F_p[v_n,v_n^{-1}],\qquad |v_n|=2(p^n-1).
$$
Here multiplication by $v_n$ is invertible and implements periodicity. Every nonzero homogeneous coefficient is invertible, so this is a graded field, although it is not a field as an ungraded ring.<Morava E-Theory and Morava K-Theory by Jacob Lurie>[Lecture 22, Definition 5 and the ensuing coefficient computation, page 3]{ref}
For $p=2,n=1$, the period is two: the coefficient group is $\mathbb F_2$ in each even degree and zero in odd degrees. Distinct heights supply different tests for <topological spectra>; vanishing of $K(n)_*(X)$ means that this particular probe misses $X$, not that $X$ is necessarily trivial. This is the organizing idea of <chromatic homotopy theory>.
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