On a theorem of Benard. (Q985048)

The theorem of \textit{M. Benard} as mentioned in the title [J. Algebra 22, 374-377 (1972; Zbl 0239.20006)] is the following Theorem 1: Assume \(k\) is a finite Abelian extension of \(\mathbb{Q}\). Let \(p\) be a place of \(\mathbb{Q}\) (possibly \(p=\infty\)) and let \(v,w\) be two arbitrary places of \(k\) lying above \(p\). Then \(k_v\otimes_kA\) and \(k_w\otimes_kA\) have the same index [here \(k_v\) denotes the completion of \(k\) at \(v\), etc.; \(A= A(\chi,k)\) is the simple component of the group algebra \(k[G]\) of \(G\) over \(k\) corresponding to \(\chi\), i.e. \(\chi(A)\neq\{0\}\), \(\chi\) being extended by linearity to a character of \(k[G]\), where \(\chi\in\text{Irr}(G)\) such that \(\chi(g)\in k\) for all \(g\in G\)]. It is known that the conclusion of Theorem 1 holds also for \(k\) being a finite Galois extension of \(\mathbb{Q}\). In Bénard's paper, see the reference mentioned above, it is stated that for some prime \(\overline p\) of an algebraic number field \(K\) dividing \(p\), \(K_{\overline p}= K\mathbb{Q}_p\) holds. From this, with \(\chi\) an irreducible character of \(G\) over the algebraic closure \(\overline k\), it is derived that \(m_{K_{\overline p}}(\chi)=m_{K\mathbb{Q}_p}(\chi)\) holds for all primes \(\overline p\) of \(K\) dividing \(p\). This, however, is false as Ohmori remarks! Indeed, the composition \(K\mathbb{Q}\) cannot be defined canonically. The (first) half of the paper deals with questions like this; we omit the details. The second theme of the paper deals with work begun by the author [\textit{J. Ohmori}, Hokkaido Math. J. 33, No.\,2, 299--317 (2004; Zbl 1149.20009)] on \textit{W. Feit's} definition of the Schur index in his book [Characters of finite groups, New York--Amsterdam: W. A. Benjamin (1967; Zbl 0166.29002)]. The reader is kindly referred to that paper and this paper under review; as the results are very detailed but innovative and beautiful, we think that the paper under review is very worthwhile to study.