Higher reciprocity law and An analogue of the Grunwald--Wang theorem for the ring of polynomials over an ultra-finite field

TL;DR

This paper develops an explicit higher reciprocity law for polynomial rings over ultraproducts of finite fields, extending classical results to infinite fields and establishing an analogue of the Grunwald--Wang theorem.

Contribution

It introduces a higher reciprocity law for polynomial rings over ultraproducts of finite fields, generalizing classical reciprocity laws and proving an analogue of the Grunwald--Wang theorem.

Findings

Explicit higher reciprocity law for ultraproducts of finite fields

Extension of reciprocity law to infinite fields in any characteristic

Analogue of the Grunwald--Wang theorem for polynomial rings over ultraproducts

Abstract

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 $\mathbb{F}_q[t]$ over a finite field $\mathbb{F}_q$ that is due to Dedekind, K\"uhne, 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 $p$.