Injectivity and surjectivity of the Dress map

For a nontrivial finite Galois extension L/kL/k (where the characteristic of k is different from 2) with Galois group G  , we prove that the Dress map hL/k:A(G)→GW(k)hL/k:A(G)→GW(k) is injective if and only if L=k(α) where α   is not a sum of squares in k×k×. Furthermore, we prove that hL/khL/k is surjective if and only if k is quadratically closed in L  . As a consequence, we give strong necessary conditions for faithfulness of the Heller–Ormsby functor cL/k⁎:SHG→SHk, as well as strong necessary conditions for fullness of cL/k⁎.