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Bases normales autoduales et groupes unitaires en caractéristique 2

Let k be a field of characteristic 2 and let L/k be a finite Galois extension with Galois group G. We show the equivalence of the following two properties: (*) The group G is generated by elements of order 2 and by elements of odd order. (**) There exists an element x of L such that Tr(x) = 1 and T(x.g(x)) = 0 for every non trivial element g of G.