Local fields and extraordinary K-theory

We describe integral lifts K(L), indexed by local fields L of degree n = [L:\Q_p], of the extraordinary cohomology theories K(n), and apply the generalized character theory of Hopkins, Kuhn and Ravenel to identify K(L)(BG) \otimes \Q$, for a finite group G, as a ring of functions on a certain scheme \frak C_LG \'etale over L, whose points are conjugacy classes of homomorphisms from the valuation ring of L to G. When L is \Q_p this specializes to a classical theorem of Artin and Atiyah.