Eichler's trace formula for traces of the Brandt-Eichler matrices is proved for arbitrary totally definite orders in central simple algebras of prime index over global fields. A formula for type numbers of such orders is proved as an application.
The paper was motivated by Kovacs’ paper (1973), Isaacs’ paper (1980) and a recent paper, due to Brešar et al. (2018), concerning Skolem-Noether algebras. Let be a unital commutative ring, not necessarily a field. Given a unital -algebra , where is contained in the center of , , the goal of this paper is to study the question: when can a homomorphism be extended to an inner automorphism of ? As an application of main results presented in the paper, it is proved that if is a semilocal...