Jacobi sum

In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums J(χ, ψ) for Dirichlet characters χ, ψ modulo a prime number p, defined by

J

(

χ

,

ψ

)

=

∑

χ

(

a

)

ψ

(

1

−

a

)

,

{\displaystyle J(\chi ,\psi )=\sum \chi (a)\psi (1-a)\,,}

where the summation runs over all residues a = 2, 3, ..., p − 1 mod p (for which neither a nor 1 − a is 0). Jacobi sums are the analogues for finite fields of the beta function. Such sums were introduced by C. G. J. Jacobi early in the nineteenth century in connection with the theory of cyclotomy. Jacobi sums J can be factored generically into products of powers of Gauss sums g. For example, when the character χψ is nontrivial,

J

(

χ

,

ψ

)

=

g

(

χ

)

g

(

ψ

)

g

(

χ

ψ

)

,

{\displaystyle J(\chi ,\psi )={\frac {g(\chi )g(\psi )}{g(\chi \psi )}}\,,}

analogous to the formula for the beta function in terms of gamma functions. Since the nontrivial Gauss sums g have absolute value p1⁄2, it follows that J(χ, ψ) also has absolute value p1⁄2 when the characters χψ, χ, ψ are nontrivial. Jacobi sums J lie in smaller cyclotomic fields than do the nontrivial Gauss sums g. The summands of J(χ, ψ) for example involve no pth root of unity, but rather involve just values which lie in the cyclotomic field of (p − 1)th roots of unity. Like Gauss sums, Jacobi sums have known prime ideal factorisations in their cyclotomic fields; see Stickelberger's theorem.