  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 $n \ge 1$.
