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Gauss sum (Definition)

Let $ p$ be a prime. Let $ \chi$ be any multiplicative group character on $ \mathbb{Z}/p\mathbb{Z}$ (that is, any group homomorphism of multiplicative groups $ (\mathbb{Z}/p\mathbb{Z})^\times \to \mathbb{C}^\times$). For any $ a \in \mathbb{Z}/p\mathbb{Z}$, the complex number

$\displaystyle g_a(\chi) := \sum_{t \in \mathbb{Z}/p\mathbb{Z}} \chi(t) e^{2 \pi i a t/p} $
is called a Gauss sum on $ \mathbb{Z}/p\mathbb{Z}$ associated to $ \chi$.

In general, the equation $ g_a(\chi) = \chi(a^{-1}) g_1(\chi)$ (for nontrivial $ a$ and $ \chi$) reduces the computation of general Gauss sums to that of $ g_1(\chi)$. The absolute value of $ g_1(\chi)$ is always $ \sqrt{p}$ as long as $ \chi$ is nontrivial, and if $ \chi$ is a quadratic character (that is, $ \chi(t)$ is the Legendre symbol $ \left(\frac{t}{p}\right)$), then the value of the Gauss sum is known to be

\begin{displaymath} g_1(\chi) = \begin{cases} \sqrt{p}, & p \equiv 1 \pmod{4}, \ i \sqrt{p}, & p \equiv 3 \pmod{4}. \end{cases}\end{displaymath}

Bibliography

1
Kenneth Ireland & Michael Rosen, A Classical Introduction to Modern Number Theory, Second Edition, Springer-Verlag, 1990.



"Gauss sum" is owned by djao.
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See Also: Kloosterman sum


Attachments:
derivation of Gauss sum up to a sign (Derivation) by bbukh
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Cross-references: Legendre symbol, absolute value, equation, complex number, group homomorphism, character, multiplicative group, prime
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This is version 4 of Gauss sum, born on 2002-06-22, modified 2003-10-18.
Object id is 3126, canonical name is GaussSum.
Accessed 9486 times total.

Classification:
AMS MSC11L05 (Number theory :: Exponential sums and character sums :: Gauss and Kloosterman sums; generalizations)

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