On geometry of the ring of algebraic integers

Igor Nikolaev

Published 2018-04-05, updated 2018-09-13Version 2

We study geometry of the ring of algebraic integers $O_K$ of a number field $K$. Namely, it is proved that the inclusion $\mathbf{Z}\subset O_K$ defines a covering of the Riemann sphere $\mathbf{C}P^1$ ramified over three points $\{0,1,\infty\}$. Our approach is based on the notion of a Serre $C^*$-algebra. As an application, a new short proof of the Belyi Theorem is given.