Polynomial-time algorithms in algebraic number theory

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 $\mathcal{O}$.