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**Forms in odd degree extensions and self-dual normal bases.**
*(English)*
Zbl 0729.12006

Let A be a finite dimensional algebra over a field K, with K-linear involution J: \(A\to A\). The involution can be extended to an involution \(J_ L\) of \(A_ L=A\otimes_ KL,\) for any field extension L/K. Let \(N(L)=\{a\in A_ L:\;aJ_ L(a)=1\}\) and let \(K_ s\) be a separable closure of K. Let \(H^ 1(K,N)\) denote the Galois cohomology set \(H^ 1(Gal(K_ s/K), N(K_ s)).\) The authors prove the following theorem, which answers a question posed by J.-P. Serre [Cohomologie galoisienne des groupes algébriques linéaires, Colloque Théor. Groupes Algébr., Bruxelles 1962, 53-68 (1962; Zbl 0145.175)]: If L is a finite extension of odd degree of K, then the canonical map \(H^ 1(K,N) \to H^ 1(L,N)\) is injective. This result has several applications. One of them is concerned with systems of bilinear forms. If two systems of bilinear forms over K become isomorphic over an extension of odd degree, then they are already isomorphic over K.

A basis \(\{e_ 1,...,e_ n\}\) of the K-vector space L is said to be self-dual if \(Tr_{L/K}(e_ ie_ j)=\delta_{ij}.\) Let \(G=Gal(L/K)\). If the set \(\{g(x): g\in G\}\) is a basis of L over K (for a fixed x), then it is called a normal basis. The authors show that any finite Galois extension of odd degree has a self-dual normal basis. For finite abelian extensions, the converse statement is also true. These results are valid for fields of characteristic not 2. If char K\(=2\) and L/K is a finite abelian extension, then L has a self-dual normal basis if and only if the exponent of G is not divisible by 4. [Cf. also the first author, Indag. Math. 51, 379-383 (1989; Zbl 0709.12004)).

A basis \(\{e_ 1,...,e_ n\}\) of the K-vector space L is said to be self-dual if \(Tr_{L/K}(e_ ie_ j)=\delta_{ij}.\) Let \(G=Gal(L/K)\). If the set \(\{g(x): g\in G\}\) is a basis of L over K (for a fixed x), then it is called a normal basis. The authors show that any finite Galois extension of odd degree has a self-dual normal basis. For finite abelian extensions, the converse statement is also true. These results are valid for fields of characteristic not 2. If char K\(=2\) and L/K is a finite abelian extension, then L has a self-dual normal basis if and only if the exponent of G is not divisible by 4. [Cf. also the first author, Indag. Math. 51, 379-383 (1989; Zbl 0709.12004)).

Reviewer: M.Kula (Katowice)