On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914400782254080 |
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| author | Wu, Hai-Liang Wang, Li-Yuan Pan, Hao |
| author_facet | Wu, Hai-Liang Wang, Li-Yuan Pan, Hao |
| contents | Inspired by Weil's classical result on the zeta function of projective Fermat curve defined over a finite field, in this paper, we investigate some arithmetic properties of the cyclotomic matrix
$$\det\left[J_p(χ^{ki},χ^{kj})\right]_{1\le i,j\le n-1},$$
where $p\ge3$ is a prime, $1\le k<p-1$ is a divisor of $p-1$ with $p-1=kn$, $χ$ is a generator of the group of all multiplicative characters of $\mathbb{F}_p$ and $J_p(χ^{ki},χ^{kj})$ is the Jacobi sum. For example, let $ζ_p\in\mathbb{C}$ be a primitive $p$-th root of unity and $P_k(T)$ be the minimal polynomial of the algebraic integer
$$θ_k=\sum_{x\in\mathbb{F}_p,x^k=1}ζ_p^x$$
over $\mathbb{Q}$. Then we prove that
$$\det \left[J_p(χ^{ki},χ^{kj})\right]_{1\le i,j\le n-1}=(-1)^{\frac{(k+1)(n^2-n)}{2}}\cdot n^{n-2}\cdot x_p(k),$$
where $x_p(k)$ is the coefficient of $T$ in $P_k(T)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14316 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums Wu, Hai-Liang Wang, Li-Yuan Pan, Hao Number Theory Inspired by Weil's classical result on the zeta function of projective Fermat curve defined over a finite field, in this paper, we investigate some arithmetic properties of the cyclotomic matrix $$\det\left[J_p(χ^{ki},χ^{kj})\right]_{1\le i,j\le n-1},$$ where $p\ge3$ is a prime, $1\le k<p-1$ is a divisor of $p-1$ with $p-1=kn$, $χ$ is a generator of the group of all multiplicative characters of $\mathbb{F}_p$ and $J_p(χ^{ki},χ^{kj})$ is the Jacobi sum. For example, let $ζ_p\in\mathbb{C}$ be a primitive $p$-th root of unity and $P_k(T)$ be the minimal polynomial of the algebraic integer $$θ_k=\sum_{x\in\mathbb{F}_p,x^k=1}ζ_p^x$$ over $\mathbb{Q}$. Then we prove that $$\det \left[J_p(χ^{ki},χ^{kj})\right]_{1\le i,j\le n-1}=(-1)^{\frac{(k+1)(n^2-n)}{2}}\cdot n^{n-2}\cdot x_p(k),$$ where $x_p(k)$ is the coefficient of $T$ in $P_k(T)$. |
| title | On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.14316 |