On the unramified extension of an arithmetic function field in several variables

TL;DR

This paper explores unramified extensions of arithmetic function fields in several variables, linking scheme theory with algebraic number theory, and demonstrates triviality of certain unramified extensions and fundamental groups.

Contribution

It introduces a scheme-theoretic notion of unramified extensions for arithmetic function fields in multiple variables, connecting to class field theory and fundamental groups.

Findings

Purely transcendental extensions over the rational field have trivial unramified extensions.

The affine scheme over rings of integers in several variables has a trivial étale fundamental group.

Operations on unramified extensions such as base changes and subfields are established.

Abstract

In this paper we will give a scheme-theoretic discussion on the unramified extensions of an arithmetic function field in several variables. The notion of unramified discussed here is parallel to that in algebraic number theory and for the case of classical varieties, coincides with that in Lang's theory of unramified class fields of a function field in several variables. It is twofold for us to introduce the notion of unramified. One is for the computation of the \'{e}tale fundamental group of an arithmetic scheme; the other is for an ideal-theoretic theory of unramified class fields over an arithmetic function field in several variables. Fortunately, in the paper we will also have operations on unramified extensions such as base changes, composites, subfields, transitivity, etc. It will be proved that a purely transcendental extension over the rational field has a trivial unramified extension. As an application, it will be seen that the affine scheme of a ring over the ring of integers in several variables has a trivial \'{e}tale fundamental group.