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

Abstract

Let e ≥ 2 be an integer, p r be a prime power with p r ≡ 1 (mod e) and ηr(i) be Gaussian periods of degree e for Fpr . 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 ηr(0), . . . , ηr(e − 1) which are defined by F. Thaine. We also give some examples of the explicit lifts for prime degree e with 3 ≤ e ≤ 23 which also illustrate relations among lifts of Jacobi sums, Gaussian periods and multiplication matrices of Gaussian periods.