Davenport and Hasse's theorems and lifts of multiplication matrices of Gaussian periods

Abstract

Let $e\ge 2$ be an integer, $p^r$ be a prime power with $p^r\equiv 1(\mathrm{mod}e)$ and ${\eta }_r(i)$ be Gaussian periods of degree $e$ for ${\mathbb{F}}_{p^r}$. By the dual form of Davenport and Hasse's lifting theorem on Gauss sums, we establish lifts of the multiplication matrices of the Gaussian periods ${\eta }_r(0),\dots ,{\eta }_r(e-1)$ which are defined by F. Thaine. We also give some examples of the explicit lifts for prime degree $e$ with $3\le e\le 23$ which also illustrate relations among lifts of Jacobi sums, Gaussian periods and multiplication matrices of Gaussian periods.

arXiv commentTo appear in Finite Fields Appl., 28 pages