Norm relations and computational problems in number fields

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.