Hasse–Davenport relation

The Hasse–Davenport relation, named after Helmut Hasse and Harold Davenport, is an equality in number theory relating gauss sums. One of its uses is in proving results regarding the zeta function of an algebraic variety.

Statement of theorem

Let "F" be a finite field with "q" elements, and "F"_s be the the field such that ["F"_s:"F"] = "s", that is "s" is the dimension of the vector space "F"_s over "F".

Let $alpha$ be an element of $F\_s$.

Let $chi$ be a multiplicative character from "F" to the complex numbers.

Let $N\_\{F\_s/F\}(alpha)$ be the norm from $F\_s$ to $F$ defined by:$N\_\{F\_s/F\}(alpha):=alphacdotalpha^qcdotsalpha^\{q^\{s-1.,$

Let $chi\text{'}$ be the composition of $chi$ with the norm from "F"_s to "F", that is:$chi\text{'}(alpha):=chi(N\_\{F\_s/F\}(alpha))$

Let $g(chi)$ be the gauss sum.

Then the Hasse–Davenport relation says that:$(-1)^scdot\; g(chi)^s=-g(chi\text{'}).$

References

#cite book
last = Ireland
first = Kenneth
coauthors = Michael Rosen
title = A Classical Introduction to Modern Number Theory
publisher = Springer
date = 1990
pages = 158-162
isbn = 0-387-97329-X