2023/09/06 by Dong Quan Ngoc Nguyen, Nguyen, Dong Quan Ngoc
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2309.02734
openalex publication_date 2023/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we establish an explicit higher reciprocity law for the polynomial ring over a nonprincipal ultraproduct of finite fields. Such an ultraproduct can be taken over the same finite field, which allows to recover the classical higher reciprocity law for the polynomial ring \mathbbFq[t] over a finite field \mathbbFq that is due to Dedekind, Kühne, Artin, and Schmidt. On the other hand, when the ultraproduct is taken over finite fields of unbounded cardinalities, we obtain an explicit higher reciprocity law for the polynomial ring over an infinite field in both characteristics 0 and p >0 for some prime p. We then use the higher reciprocity law to prove an analogue of the Grunwald--Wang theorem for such a polynomial ring in both characteristics 0 and p > 0 for some prime p.