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.

[edit] 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 $α$ be an element of $F s$ .

Let $χ$ 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

$N_{F_s/F}(\alpha):=\alpha\cdot\alpha^q\cdots\alpha^{q^{s-1}}.\,$

Let $χ'$ be the composition of $χ$ with the norm from F_s to F, that is

$\chi'(\alpha):=\chi(N_{F_s/F}(\alpha))$

Let $g (χ)$ be the gauss sum.

Then the Hasse–Davenport relation says that

$(-1)^s\cdot g(\chi)^s=-g(\chi').$

Ireland, Kenneth; Michael Rosen (1990). A Classical Introduction to Modern Number Theory. Springer, 158-162. ISBN 0-387-97329-X.