Galois algebras, Hasse principle and induction-restriction methods
Let \(L/k\) be a Galois field extension with group \(G\). It is well known that \(L/k\) has a \textit{normal basis}, i.e., a \(k\)-vector space basis which consists of a single \(G\)-orbit in \(L\). Assume \(\mathrm{char}(k) \neq 2\). To \(L/k\) we can associate its trace form \(q_L: L\times L\to k\), defined by \(q_L(x,y)=\mathrm{Tr}_{L/k}(xy)\). It is a \(G\)-quadratic form on \(L\), i.e., satisfies \(q_L(gx,gy)=q_L(x,y)\) for all \(g\in G\) and \(x,y\in L\). The question arises whether \(L/k\) has a normal basis which satisfies \(q_L(gx,hx)=\delta_{g,h}\) for all \(g,h\in G\) and \(x\in L\). Such a normal basis is called \textit{self-dual}. This problem has been considered in [\textit{E. Bayer-Fluckiger} and \textit{H.W. Lenstra, jun.}, Am. J. Math. 112, No. 3, 359--373 (1990; Zbl 0729.12006)], where a positive answer is given in case \([L:k]\) is odd, and in [\textit{E. Bayer-Fluckiger} and \textit{J.-P. Serre}, Am. J. Math. 116, No. 1, 1--64 (1994; Zbl 0804.12004)], which contains partial results in case \([L:k]\) is even. The focus of the paper under review is on the following Hasse principle type of question for global fields \(k\): Suppose that \(L\) and \(L'\) are \(G\)-Galois algebras (e.g. \(G\)-Galois field extensions) over \(k\). Assume that for all places \(\nu\) of \(k\) the \(G\)-forms \(q_{L_\nu}\) and \(q_{L'_\nu}\) are isomorphic over the completion \(k_\nu\). Are the \(G\)-forms \(q_L\) and \(q_{L'}\) isomorphic over \(k\)? Note that \(L/k\) has a self-dual normal basis if and only if \(q_L\) is isomorphic as a \(G\)-quadratic form to the unit form \(u=q_{k[G]}\) on the group algebra \(k[G]\), determined by \(u(g,h)=\delta_{g,h}\). A key ingredient for analyzing this question is the \textit{odd determinant property} of a finite group \(G\) introduced in this paper. Choose a \(2\)-Sylow subgroup \(S\) of \(G\) and let \(X\) be the free abelian group of \(\mathbb{Z}\)-valued functions on \(S\) invariant under conjugation by the normalizer \(N=N_{G}(S)\). Consider the endomorphism \(\Phi: X\to X\) of \(X\) given by induction from \(S\) to \(G\) followed by restriction from \(G\) to \(S\), defined as for characters of \(S\). The group \(G\) is said to have the odd determinant property if \(\det(\Phi) \in \mathbb{Z}\) is odd. The main result of the paper is the following result: Let \(L\) and \(L'\) be two \(G\)-Galois algebras over the global field \(k\) of characteristic \(\neq 2\). In case \(\mathrm{char}(k)=0\) assume that \(G\) has the odd determinant property. Then the \(G\)-quadratic forms \(q_L\) and \(q_{L'}\) are isomorphic if and only if they are isomorphic over every completion of \(k\). In case \(L'=k[G]\) this means that \(L\) has a self-dual normal basis if and only if it has a self-dual normal basis over every completion of \(k\). The odd determinant property is shown to be satisfied if for the chosen \(2\)-Sylow subgroup \(S\) its normalizer \(N=N_G(S)\) controls the fusion of \(S\) in \(G\). This means that for all subsets \(T\) in \(S\) and \(g\in G\) with \(gTg^{-1}\subseteq S\) there exists \(n\in N\) with \(gtg^{-1}=ntn^{-1}\) for all \(t\in T\). This happens in particular if \(S\) is abelian (by a theorem of Burnside) or if \(S\) is normal in \(G\). Further examples are given in [\textit{J. Thévenaz}, Expo. Math. 11, No. 4, 359--363 (1993; Zbl 0809.20013)]. In the paper it is shown that the matrix of \(\Phi: X\to X\) in its canonical basis is diagonal with odd entries on the diagonal, from which the odd determinant property follows. The largest part of the paper is devoted to the proof the following intermediate result: Suppose that \(G\) has the odd determinant property. Let \((V_1,q_1)\) and \((V_2,q_2)\) be two \(S\)-quadratic spaces. Suppose that the induced \(G\)-quadratic spaces \(\mathrm{Ind}_S^G (V_1,q_1)\) and \(\mathrm{Ind}_S^G (V_2,q_2)\) (as defined in E. Bayer-Fluckiger and J.-P. Serre [Zbl 0804.12004]) become isomorphic after restriction to \(S\). Then they are isomorphic.
scientific article; zbMATH DE number 5987891