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 double-struckQfalse[Gfalse]$\mathbb {Q}[G]$. 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.

Applications of representation theory and of explicit units to Leopoldt's conjecture

“…By using so-called 'generalised useful idempotent relations' (see §4.1) and a key result of [BFHP22], we prove the following result in §4.6.…”

An algorithm for solving the principal ideal problem with subfields

“…In contrast, in this paper, we use norm relations to solve the PIP without having to compute S-unit groups. This results in a significant practical speed-up over the direct application of [14]. In addition, we are able to solve the PIP in fields of degree significantly larger than the previous state of the art.…”

“…In another direction, the method of [3] was adjusted by Lesavourey, Plantard and Susilo [32] to the case of multicubic fields. Recent work from Biasse, Fieker, Hofmann and Page [14] generalized this concept of norm relation involving subfields to a large variety of number fields (essentially consisting of fields whose Galois group is "far from" being cyclic). Among other computational tasks, they showed how to leverage these relations to compute S-unit groups and ideal class groups recursively using subfields.…”

“…Our contribution We use the norm relations construction technique introduced by Biasse, Fieker, Hofmann and Page [14] to efficiently solve the PIP recursively in non-cyclic number fields. This framework includes the prior work of [3] on multiquadratics and extends it to a significantly larger variety of fields.…”

“…This framework includes the prior work of [3] on multiquadratics and extends it to a significantly larger variety of fields. The prior work of [14] allows the recursive computation of S-unit groups from subfields, which in turn can be used to solve the PIP [9]. In contrast, in this paper, we use norm relations to solve the PIP without having to compute S-unit groups.…”