A Gauss sum is a particular kind of finite sum of roots of unity, typically formed by summing a multiplicative character of a finite field weighted by an additive character such as a root of unity raised to a power depending on the summation index. First studied by Carl Friedrich Gauss in connection with quadratic residues, these sums have a remarkably simple absolute value despite their complex-looking definition, a fact Gauss proved for the classical quadratic case. Gauss sums appear throughout number theory in the study of quadratic reciprocity, L-functions and the functional equations of various zeta and L-series.
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In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically G ( χ):= G ( χ, ψ) = ∑ χ ( r) ⋅ ψ ( r) {\displaystyle G(\chi):=G(\chi,\psi)=\sum \chi (r)\cdot \psi (r)} where the sum is over elements r of some finite commutative ring R, ψ is a group homomorphism of the additive group R+ into the unit circle
In Branch: Algebraic Number Theory, Lead sentence
In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically G ( χ ) :=
Attributed To: Carl Friedrich Gauss, Lead
The case originally considered by Carl Friedrich Gauss was the quadratic Gauss sum, for R the field of residues modulo a prime number p, and χ the Legendre symbol
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