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

TL;DR

This paper extends Davenport and Hasse's theorems to lift multiplication matrices of Gaussian periods over finite fields, providing explicit examples and exploring their relations with Jacobi sums.

Contribution

It introduces a method to lift multiplication matrices of Gaussian periods using the dual form of Davenport and Hasse's lifting theorem, with explicit examples for prime degrees.

Findings

Lifts of multiplication matrices are explicitly constructed for certain prime degrees.

Relations among lifts of Jacobi sums, Gaussian periods, and multiplication matrices are demonstrated.

Examples for prime degrees 3 to 23 illustrate the theoretical results.

Abstract

Let $e \geq 2$ be an integer, $p^r$ be a prime power with $p^r \equiv 1\ ({\rm mod}\ e)$ 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 $\eta_r(0),\ldots,\eta_r(e-1)$ which are defined by F. Thaine. We also give some examples of the explicit lifts for prime degree $e$ with $3\leq e\leq 23$ which also illustrate relations among lifts of Jacobi sums, Gaussian periods and multiplication matrices of Gaussian periods.