Quotient rings and fields of integers from the metric geometry point of view

Abstract

The theory of Gromov-Hausdorff convergence is applied to sequences of quotient rings of integers, leading to the existence of limit rings as the Gromov-Hausdorff limits. It is also shown that such {\it limit rings} can be endowed with an order relation, although they are not ordered fields. When the construction restricts to quotients $\mathbb{Z}/p\mathbb{Z}$ with $p$ prime, the limits are fields. The relation of those limits with the field of the real numbers $\mathbb{R}$ is discussed, showing that the limit fields are dense in $\mathbb{R}$ but they cannot be identified with $\mathbb{R}$ or with the rational field $\mathbb{Q}$, at least when $\mathbb{R}$ and $\mathbb{Q}$ are endowed with the usual order relations and metric structures, neither they can be identified with the $p$-adic number system. It is also shown that the limit rings and fields have an ordinal larger than the reals and that they are endowed with an Archimedean absolute value function.