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Let  p  > 2 be prime and let  n, m  ∈ ℕ be given. For cyclic extensions  E / F  of degree  p   n   that contain a primitive  p th root of unity, we show that the associated   p  [Gal( E / F )]‐modules  H   m  ( G   E  ,  μ   p  ) have a sparse decomposition. When  E / F  is additionally a subextension of a cyclic, degree  p   n +1  extension  E ′/ F , we give a more refined   p  [Gal( E / F )]‐decomposition of  H   m  ( G   E  ,  μ   p  ).

Galois module structure of Galois cohomology for embeddable cyclic extensions of degree p n