Noetherian equations for matrices
Let \(L\) be a finite Galois extension of the field \(K\) and denote by \(G\) the Galois group \(\text{Gal}(L/K)\). The author investigates two kinds of ``Noetherian equations''. The first kind is the set of solutions of the matrix equation \[ X_{\sigma \tau}=X_\sigma^\tau X_\tau, \] where \(\sigma,\tau\) are in \(G\), \(X_\sigma\) is in \(GL_n(L)\) for a given natural number \(n\), and \(X_\sigma^\tau \) is obtained from \(X_\sigma\) by applying \(\tau\) to all entries of \(X_\sigma\). The map which associates \(\sigma\) to \(X_\sigma\) is called by the author a crossed representation of \(G\) over \(L\). The second kind is the set of solutions of \[ X_{\sigma \tau} f_{\sigma, \tau}=X_\sigma^\tau X_\tau, \] where \(X_\sigma\) is in \(GL_n(L)\) and \(f_{\sigma, \tau}\) is in \(L^\times\) (the invertible elements of \(L\)). In fact it is the previous equation with the solutions in \(PGL_n(L)\), and the author calls the map which associates \(\sigma\) to \(X_\sigma\), crossed-projective representation of \(G\) over \(L\). We can see that \(f_{\sigma,\tau}\) is in fact a 2-cocycle over \(G\) with coefficient in \(L^\times\). With a 2-cocycle \(f_{\sigma,\tau}\), we can form the crossed-product of \(L\) and \(G\) over \(K\) relative to \(f\), denoted by \((L,G,f)\), which is a central simple algebra over \(K\). Then the author shows (Corollary 1, p. 318) that all the solutions of the first equation are given by \(X_\sigma=Q^{-\sigma} Q\), where \(Q\) ranges over the elements of \(GL_n(L)\), and he shows (Proposition 4, p. 320) that there is a correspondence, up to equivalence, between 2-cocycles and solutions of the second equation. He also shows (Proposition 5, p. 320) that the minimal degree of a solution of the second equation (i.e. the smallest \(n\) such that \(X_\sigma\) is in \(GL_n(L)\)) coincides with the Schur index of the algebra \((L,G,f)\). The author also gives some properties in the case of cyclic extensions. Rephrased in the cohomological language, the main results of the author say that \[ H^1(G,GL_n(L))=0 \] and that we have a bijection, induced by the ``crossed product construction'', between \(H^1(G,PGL_n(L))\) and \(H^2(G,L^\times)\), where here \(n\) is the degree of \(L\) over \(K\). And these results are known as consequences of the modern version of Hilbert's Theorem 90 and some tools involving Galois cohomology. The reader can find comprehensive proofs of the previous statements in the chapter 7 ``Galois cohomology'', of the book of \textit{M.-A. Knus, A. Merkurjev, M. Rost} and \textit{J.-P. Tignol} [The book of involutions. With a preface by J. Tits. Colloquium Publications. American Mathematical Society. 44. Providence, RI (1998)]. It appears, in fact, that the author has re-discovered results of \textit{A. Speiser} [``Zahlentheoretische Sätze aus der Gruppentheorie'', Math. Z. 5, 1-6 (1919; JFM 47.0092.01)] and \textit{I. Schur} [``Einige Bemerkungen zu der vorstehenden Arbeit des Herrn A. Speiser'', Math. Z. 5, 7-10 (1919; JFM 47.0092.02)]. Indeed A. Speiser (Satz 1, p. 2, op. cit.) has described the set of solution ot the first equation and I. Schur (Satz I, p. 8, op. cit.), using the notion of factor system, has described the set of solutions of the second equations, and has also shown (Satz III, p. 10, op. cit.), like the author, that, given a factor system (i.e. a 2-cocycle), there is, up to equivalence, only one irreducible solution. Notice also that in the paper under review, several references to Emmy Noether could be misleading and I would recommend as a complementary reading the introduction, by N. Jacobson, to the Collected Works of \textit{E. Noether} [Gesammelte Abhandlungen. Collected papers. Springer-Verlag (1983; Zbl 0504.01035)].
scientific article; zbMATH DE number 1218155