  We construct Galois extensions of the T(n)-local sphere, lifting all finite
abelian Galois extensions of the K(n)-local sphere. This is achieved by
realizing them as higher semiadditive analogues of cyclotomic extensions.
Combining this with a general form of Kummer theory, we lift certain elements
from the K(n)-local Picard group to the T(n)-local Picard group.
