Über die gruppentheoretische Struktur der Relationen zwischen Relativnormabbildungen in endlichen Galoisschen Körpererweiterungen

Abstract

We define and study the module (over rational integers) of relative norms in galois field extensions in a purely group theoretical manner. We can compute a basis of this module from the lattice of subgroups by means of the Möbius function. Its rank is the number of noncyclic subgroups. The module depends functorially on the group. The module of a factor group is naturially imbedded in the module of the group. We prove some other structure results of this type.A norm relation relates those subfields which have nonzero coefficient in this relation, the other fields being eliminated.Using deep group theoretical results by Zassenhaus and Suzuki we can give an explicit description of all those groups for which the trivial subgroup is eliminated in any norm relation. (This means for example that we know the group types of all those galois number fields in which norm relations do not yield an estimate of the divisor class group exponent by means of similar exponents of subfields.) The list is somewhat unexpected, it includes special linear groups of dimension 2 over galois fields of Fermat prime characteristic and certain metacyclic groups.