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In this paper, we establish an explicit higher reciprocity law for thepolynomial ring over a nonprincipal ultraproduct of finite fields. Such anultraproduct can be taken over the same finite field, which allows to recoverthe 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, andSchmidt. On the other hand, when the ultraproduct is taken over finite fieldsof unbounded cardinalities, we obtain an explicit higher reciprocity law forthe 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 ananalogue of the Grunwald--Wang theorem for such a polynomial ring in bothcharacteristics $0$ and $p > 0$ for some prime $p$.

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