Ultra-Galois theory and an analogue of the Kronecker--Weber theorem for rational function fields over ultra-finite fields

Abstract

In the first part of this paper, we develop a general framework that permits a comparison between explicit class field theories for a family of rational function fields $\mathbb{F}_s(t)$ over arbitrary constant fields $\mathbb{F}_s$ and explicit class field theory for the rational function field $\mathfrak{K}(t)$ over the nonprincipal ultraproduct $\mathfrak{K}$ of the constant fields $\mathbb{F}_s$. Under an additional assumption that the constant fields $\mathbb{F}_s$ are perfect procyclic fields, we prove a correspondence between ramifications of primes $P$ in $\mathfrak{K}(t)$ and ramifications of primes $P_s$ in $\mathbb{F}_s(t)$, where the $P_s$ are primes in $\mathbb{F}_s(t)$ whose nonprincipal ultraproduct coincides with $P$. In the second part of the paper, we are mainly concerned with rational function fields over a large class of fields, called $n$-th level ultra-finite fields that are a generalization of finite fields. At the $0$-th level, ultra-finite fields are simply finite fields, and for an arbitrary positive integer $n$, an $n$-th level ultra-finite field is inductively defined as a nonprincipal ultraproduct of $(n - 1)$-th level ultra-finite fields. We develop an analogue of cyclotomic function fields for rational function fields over $n$-th level ultra-finite fields that generalize the works of Carlitz and Hayes for rational function fields over finite fields such that these cyclotomic function fields are in complete analogy with the classical cyclotomic fields $\mathbb{Q}(\zeta)$ of the rationals $\mathbb{Q}$. The main result in the second part of the paper is an analogue of the Kronecker--Weber theorem for rational function fields over $n$-th level ultra-finite fields that explicitly describes, from a model-theoretic viewpoint, the maximal abelian extension of the rational function field over a given $n$-th level ultra-finite field for all $n \ge 1$.