Here we shall apply the general theory, developed so far, to group rings and add further results valid specifically for these. The involution on the group ring FГ of a finite group Г over a field F is always the F-linear map FГ → FГ which takes every group element γ into γ−1. We shall give briefly a translation of the language of representations of algebras to that of representations of groups and of group characters, both in the linear context where it is well known, as well as in the Hermitian one. The principal new property which makes its appearance is that of virtual characters to form a ring rather than just an additive group, and of certain rings of characters to act on classgroups and on Hom groups in a consistent manner. This is also true in the Hermitian theory. In principle one could consider more general co-algebras rather than group rings, but we shall not do so here. Our theorems, in IV, on change of orders are now applied in the context both of change of group, and of change of basefield, and then lead to structures of Frobenius modules.
KeywordsGalois Group Number Field Prime Divisor Group Ring Maximal Order
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