On an Obstruction to the Hasse Norm Principle and the Equality of Norm Groups of Algebraic Number Fields

Abstract

LetL/kandT/kbe finite extensions of algebraic number fields. In the present work we introduce the factor group ofk*L/kJLNT/kJTby (k*T/kJT)NL/kL*J LandJTare the idele groups ofLandT, respectively. The main theorem shows that the computation of this factor group can be reduced to the computation in finite group theory, and the computation with Galois groups of local extensions at a finite number of primes of the base fieldk. We then apply the main theorem to establish a number of interesting results on the equality of norm groups as subgroups of the multiplicative group ofk. In particular, we obtain new results on solitary non-Galois extensions of algebraic number fields. © 1999 Academic Press.