Reciprocity for Jacobi sums (Q687518)

Let \(\ell\) be an odd prime number and \(p,q\) be two prime numbers \(\equiv 1 \pmod \ell\). Let \(P,Q\) be two prime ideals of \(\mathbb{Z} [I_\ell]\) dividing \(p\) and \(q\), respectively. Let \(\chi_P\) (resp. \(\chi_Q)\) be the multiplicative character of order \(\ell\) on \(\mathbb{F}_p\) (resp. \(\mathbb{F}_q)\) induced by the inverse of the \(\ell\)-th power residue symbol, over \(K\), modulo \(P\) (resp. \(Q)\). Then the following \(\ell\)-th power reciprocity relation between Jacobi sums holds: \[ \left( \frac{J(\chi^a_P, \chi^b_P)}{J(\chi^c_Q, \chi^d_Q)} \right) = \left( \frac{J(\chi^c_Q, \chi^d_Q)} {J(X^a_P, \chi^b_P)} \right). \] \(a,b,c,d\) are integers such that \(\ell \nmid abcd\).