Algorithms in Algebraic Number Theory

TLDR

The paper focuses on mathematically interesting aspects of algorithmic algebraic number theory, largely ignoring practical issues. The paper aims to survey fundamental problems in algorithmic algebraic number theory, review current progress, and highlight remaining challenges while illustrating how algorithmic study deepens understanding and sparks curiosity about algebraic number fields. The discussion focuses on three key algorithmic tasks: determining Galois groups, computing the ring of integers of an algebraic number field, and calculating its unit group and class group.

Abstract

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers.