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Aug 5, 2011 at 14:04 comment added Geoff Robinson @David: Well it isn't really cheating, but relies on having an extra $p$-th root of unity in the unramified case when $p = 2$.
Aug 5, 2011 at 13:59 comment added David E Speyer How is that cheating? You've solved the problem (for $p=2$).
Aug 5, 2011 at 13:44 comment added Frieder Ladisch Great idea. Even more simply, let $A$ be any matrix of odd dimension and order $2^r$, and take $B=-A$.
Aug 5, 2011 at 13:28 history edited Geoff Robinson CC BY-SA 3.0
typo; added 2 characters in body
Aug 5, 2011 at 13:22 history answered Geoff Robinson CC BY-SA 3.0