2024/08/04 by Dong Quan Ngoc Nguyen, Nguyen, Dong Quan Ngoc
Computer Science · #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2408.02158
openalex publication_date 2024/08/04 · openalex created_date 2025/01/17 · openalex updated_date 2026/07/28
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 \mathbbFs(t) over arbitrary constant fields \mathbbFs and explicit class field theory for the rational function field \mathfrakK(t) over the nonprincipal ultraproduct \mathfrakK of the constant fields \mathbbFs. Under an additional assumption that the constant fields \mathbbFs are perfect procyclic fields, we prove a correspondence between ramifications of primes P in \mathfrakK(t) and ramifications of primes Ps in \mathbbFs(t), where the Ps are primes in \mathbbFs(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 ℚ(ζ) of the rationals 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 ≥ 1.