Polynomial-time algorithms in algebraic number theory
Daniël M. H. van Gent
TL;DR
The notes present a comprehensive framework for polynomial-time algorithms in algebraic number theory, unifying lattice methods, coprime-basis techniques, and ideal-theoretic blowups to enable efficient computations with number rings, maximal orders, and symbols. Central to the approach are the kernel-image algorithm (via LLL), the coprime-basis factorization, and blowups that convert noninvertible ideals into invertible ones, all orchestrated to reduce hard problems to tractable linear-algebra and lattice computations. The work also establishes algorithmic equivalences linking maximal orders, nilradicals, and radical computations, underscoring deep connections between structural algebra and computability. Collectively, these methods yield polynomial-time procedures for deciding unit relations, computing Jacobi symbols in number rings, and manipulating fractional ideals, with significant implications for computational number theory and symbolic algebra systems.
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 $\mathcal{O}$.