Norm relations and computational problems in number fields

TL;DR

This work introduces norm relations in $\mathbb{Q}[G]$, connecting arithmetic invariants of a normal extension $K/F$ to those of its subfields and enabling subfield-based computations of rings of integers, $S$-units, and class groups. It provides a complete group-theoretic classification for the existence of norm relations, develops both additive and multiplicative structural consequences for $\mathcal{O}_K$, $\mathcal{O}_{K,S}^\times$, and $\mathrm{Cl}(K)$, and proves that, under GRH, these invariants admit polynomial-time reductions to subfields; it also yields new unconditional results for cyclotomic fields. The paper presents practical algorithms implemented in Hecke and Pari/GP, and demonstrates the approach with substantial numerical examples, including large non-abelian fields and cyclotomic fields up to conductor $2520$. Overall, norm relations offer a unified and effective framework for leveraging subfield data to compute central invariants in computational algebraic number theory.

Abstract

For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb Q[G]$. 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 $G$. On the algorithmic side this leads to subfield based algorithms for computing rings of integers, $S$-unit groups and class groups. For the $S$-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.

Theorems & Definitions (99)

• Theorem A
• Proposition B
• Proposition C
• Theorem D
• Theorem E
• Proposition F
• Definition 2.1
• Remark 2.2
• Example 2.3
• Example 2.4