In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These objects are named after Carl Friedrich Gauss, who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws.
Definition For an odd prime number p and an integer a, the quadratic Gauss sum g(a; p) is defined as
g ( a ; p ) = ∑ n = 0 p − 1 ζ p a n 2 , {\displaystyle g(a;p)=\sum _{n=0}^{p-1}\zeta _{p}^{an^{2}},}
where ζ p {\displaystyle \zeta _{p}} is a primitive pth root of unity, for example ζ p = exp ( 2 π i / p ) {\displaystyle \zeta _{p}=\exp(2\pi i/p)} . Equivalently, we can write this using the Legendre symbol as
g ( a ; p ) = ∑ n = 0 p − 1 ( 1 + ( n p ) ) ζ p a n . {\displaystyle g(a;p)=\sum _{n=0}^{p-1}{\big (}1+\left({\tfrac {n}{p}}\right){\big )}\,\zeta _{p}^{an}.}
For a divisible by p, and we have ζ p a n 2 = 1 {\displaystyle \zeta _{p}^{an^{2}}=1} and thus
g ( a ; p ) = p . {\displaystyle g(a;p)=p.}
For a not divisible by p, we have ∑ n = 0 p − 1 ζ p a n = 0 {\displaystyle \sum _{n=0}^{p-1}\zeta _{p}^{an}=0} , implying that
g ( a ; p ) = ∑ n = 0 p − 1 ( n p ) ζ p a n = G ( a , ( ⋅ p ) ) , {\displaystyle g(a;p)=\sum _{n=0}^{p-1}\left({\tfrac {n}{p}}\right)\,\zeta _{p}^{an}=G(a,\left({\tfrac {\cdot }{p}}\right)),}
where
G ( a , χ ) = ∑ n = 0 p − 1 χ ( n ) ζ p a n {\displaystyle G(a,\chi )=\sum _{n=0}^{p-1}\chi (n)\,\zeta _{p}^{an}}
is the Gauss sum defined for any character χ modulo p.
Properties The value of the Gauss sum is an algebraic integer in the pth cyclotomic field Q ( ζ p ) {\displaystyle \mathbb {Q} (\zeta _{p})} . The evaluation of the Gauss sum for an integer a not divisible by a prime p > 2 can be reduced to the case a = 1:
g ( a ; p ) = ( a p ) g ( 1 ; p ) . {\displaystyle g(a;p)=\left({\tfrac {a}{p}}\right)g(1;p).}
The exact value of the Gauss sum for a = 1 is given by the formula:
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