Computing class groups by induction with generalised norm relations

TL;DR

This paper generalizes norm relations in group algebras to develop an algorithm that computes class groups of large number fields by relating them to lower-degree fields, improving computational methods in algebraic number theory.

Contribution

It introduces a new generalization of norm relations in group algebras and applies them to efficiently compute class groups of complex number fields.

Findings

Successfully computed class groups of large number fields.

Established properties of generalized norm relations.

Reduced class group computations to lower-degree fields.

Abstract

We introduce a generalisation of norm relations in the group algebra Q[G], where G is a finite group. We give some properties of these relations, and use them to obtain relations between the S-unit groups of different subfields of the same Galois extension of Q, of Galois group G. Then we deduce an algorithm to compute the class groups of some number fields by reducing the problem to fields of lower degree. We compute the class groups of some large number fields.