Eisenstein sum
# Eisenstein sum

In mathematics, an Eisenstein sum is a finite sum depending on a finite field and related to a Gauss sum. Eisenstein sums were introduced by Eisenstein in 1848, named "Eisenstein sums" by Stickelberger in 1890, and rediscovered by Yamamoto in 1985, who called them relative Gauss sums.

## Definition

The Eisenstein sum is given by

    $E(\chi ,\alpha )=\sum _{Tr_{F/K}t=\alpha }\chi (t)$

where F is a finite extension of the finite field K, and χ is a character of the multiplicative group of F, and α is an element of K.

