Morava K-theory ($K(n{)}_{\ast }$)

At a prime number $p$ and height $n\ge 1$, Morava K-theory is a periodic generalized homology theory represented by a topological spectrum $K(n)$ with coefficient graded ringHere 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.^[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 $K(n{)}_{\ast }(X)$ means that this particular probe misses $X$, not that $X$ is necessarily trivial. This is the organizing idea of chromatic homotopy theory.