Norm relations and computational problems in number fields

Abstract

For a finite group G $G$ , we introduce a generalization of norm relations in the group algebra Q [ G ] $\mathbb {Q}[G]$ . We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the arithmetic invariants of the subfields of a normal extension of algebraic number fields with Galois group G $G$ . On the algorithmic side, this leads to subfield based algorithms for computing rings of integers, S $S$ -unit groups and class groups. For the S $S$ -unit group computation this yields a polynomial time reduction to the corresponding problem in subfields. We compute class groups of large number fields under GRH, and new unconditional values of class numbers of cyclotomic fields.