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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.20892 |
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| _version_ | 1866914500384391168 |
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| author | Hu, Su Wang, Enci |
| author_facet | Hu, Su Wang, Enci |
| contents | In 1991, Rousseau gave a new proof of Gauss's quadratic reciprocity by comparing two distinct coset representations of the group $(\mathbb{Z}_{p}^{*} \times \mathbb{Z}_{q}^{*}) / U$ using the Chinese Remainder Theorem, without Gauss's Lemma. In this paper, we extend Rousseau's approach to $\mathbb{F}_{q}[t]$, providing a new, elementary proof of the reciprocity law for the $d$th power residue symbol, where $d$ is any divisor of $q-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20892 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the reciprocity law in $\mathbb{F}_{q}[t]$ Hu, Su Wang, Enci Number Theory 11A15, 11R58 In 1991, Rousseau gave a new proof of Gauss's quadratic reciprocity by comparing two distinct coset representations of the group $(\mathbb{Z}_{p}^{*} \times \mathbb{Z}_{q}^{*}) / U$ using the Chinese Remainder Theorem, without Gauss's Lemma. In this paper, we extend Rousseau's approach to $\mathbb{F}_{q}[t]$, providing a new, elementary proof of the reciprocity law for the $d$th power residue symbol, where $d$ is any divisor of $q-1$. |
| title | On the reciprocity law in $\mathbb{F}_{q}[t]$ |
| topic | Number Theory 11A15, 11R58 |
| url | https://arxiv.org/abs/2604.20892 |