Quintic reciprocity

An expression for the rational inversion factor of the power residue symbol, of odd prime exponent n ≡ 1 (mod 4), is given. It is applied to the quintic case, where the resulting expression involves only a rational quadratic form representation of primes and the power residue character of Jacobi sums. A reciprocity relation for Jacobi sums is then deduced, for n = 5, and conjectured to hold for all odd prime exponents n.