of $ \Omega $ on the group algebra $ F^{\text{alg}}[N] $, as in [@BLE1999 Lemma 2.2]. As discussed above, we have $ L[N]^{G} =  F^{\text{alg}}[N]^{\Omega} $, and the action of $ \Omega $ factors through $ {\text{Gal}}{(L^{\prime}/K)} $ for some cyclotomic extension $ L^{\prime} $ of $ L $. Thus, writing $ G^{\prime} = {\text{Gal}}{(L^{\prime}/L)} $, we have $$L[N]^{G} = \left( L^{\prime}[N]^{G^{\prime}} \right)^{G},$$ where the action of $ G^{\prime} $ on $ L^{\prime}[N] $ is only on the coefficients. In the following two lemmas we suppress the details of this first step of the descent (if any), and begin with a convenient $ L $-basis on $ L[N] $ on which it is easy to follow the action of $ G $. By forming $ G $-invariant linear combinations of these basis elements we obtain a basis of $ L[N]^{G} $ corresponding to its Artin-Wedderburn decomposition. Although working with bases in this way is rather cumbersome, it has the advantage of applying uniformly, whereas studying the orbits of $ \Omega $ in $ \widehat{N} $ can split into many cases, depending upon the roots of unity contained in $ L $.

We continue with the Hopf algebras giving the structures 