 Gauss sum of a finite-field character

ID: gauss-sum-of-a-finite-field-character

Here

ψ

is a nontrivial additive character of the finite field. For nontrivial

χ

, substituting

x = t y

in the squared absolute value gives

∣ G (χ) ∣^{2} = \sum_{t \neq = 0} χ (t) \sum_{y \neq = 0} ψ ((t - 1) y) = q .

The inner sum is

q - 1

t = 1

and

- 1

otherwise; multiplicative character orthogonality completes the identity.

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