Rational and genus equivalence of forms (Q1097908)

The author continues his work on the full decomposable forms which correspond to the invertible fractional ideals of an arbitrary order D of a finite extension k of the rational field \({\mathbb{Q}}\). He proves that the set of rationally equivalent classes of such forms is a group under composition of forms, and similarly, for the set of genera. The number of rationally equivalent classes of forms in each genus of forms is a certain multiple of the index of the subgroup of global norms in the group of everywhere local norms of k/\({\mathbb{Q}}\). This generalizes the earlier results, obtained for normal extensions K/\({\mathbb{Q}}\) with the order D, invariant under its Galois group [cf. the author, J. Number Theory 14, 251-259 (1982; Zbl 0483.10019)]. Moreover, some information about the group structure induced by rational equivalence is given.