Extending Rousseau's coset comparison via the Chinese Remainder Theorem produces a new elementary proof of the d-th power reciprocity law in the polynomial ring over a finite field.
Rosen,Number theory in function fields.Graduate Texts in Mathematics, 210
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On the reciprocity law in $\mathbb{F}_{q}[t]$
Extending Rousseau's coset comparison via the Chinese Remainder Theorem produces a new elementary proof of the d-th power reciprocity law in the polynomial ring over a finite field.