Polynomial-time algorithms in algebraic number theory

TL;DR

This paper discusses the challenges and current limitations of polynomial-time algorithms in algebraic number theory, focusing on computational problems like finding the maximal order of a number field.

Contribution

It highlights the intractability of computing the maximal order and explores alternative polynomial-time approaches to address key algebraic number theory problems.

Findings

No known polynomial-time algorithm for computing the maximal order.

Focus on algorithms that work around the inaccessibility of the maximal order.

Provides insights into computational complexity in algebraic number theory.

Abstract

This document contains notes based on lectures given by Hendrik Lenstra at the PCMI summer school 2022. There are many problems in algebraic number theory which one would like to solve algorithmically, for example computation of the maximal order $\mathcal{O}$ of a number field, and the many problems that are most often stated only for $\mathcal{O}$, such as inverting ideals and unit computations. However, there is no known fast, i.e. polynomial-time, algorithm to compute $\mathcal{O}$, which we motivate by a reduction to elementary number theory. We will instead restrict to polynomial-time algorithms, and work around this inaccessibility of $\mathcal{O}$.