On the coefficients of Jacobi sums in prime cyclotomic fields期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

On the coefficients of Jacobi sums in prime cyclotomic fields

Authors:	F. Thaine

Affiliation:	Department of Mathematics and Statistics - CICMA, Concordia University, 1455, de Maisonneuve Blvd. W., Montreal, Quebec, H3G 1M8, Canada

Abstract:	Let and be prime numbers, and let be a primitive root mod . For , denote by $J_{n}$ the Jacobi sum $-sum _{k=2}^{q-1}zeta _p ^{, text{ind}_{s}(k)+n, text{ind}_{s}(1-k)}$ . We study the integers $d_{n,k}$ such that $J_{n}=sum _{k=0}^{p-1}d_{n,k}zeta _p ^{k}$ and $sum _{k=0}^{p-1}d_{n,k}=1$ . We give a list of properties that characterize these coefficients. Then we show some of their applications to the study of the arithmetic of $mathbb {Z} [zeta _p +zeta _p ^{-1}]$ , in particular to the study of Vandiver's conjecture. For , let $rho _{n}(m)$ be the number of distinct roots of $X^{n+1}-X^{n}+m$ in . We show that $d_{n,k}=f-sum _{a=0}^{f-1}rho _{n}(s^{k+pa})$ . We give closed formulas for the numbers $d_{1,k}$ and $d_{2,k}$ in terms of quadratic and cubic power residue symbols mod .

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