A note on Jacobi sums (Q1312062)

Let \(m \in \mathbb{Z}\), \(m>1\), \(\tau_ m\) a primitive \(m\)-th root of 1, \(K_ m = \mathbb{Q} (\tau_ m)\) and \(R_ m = \mathbb{Z} [\tau_ m]\). For a prime ideal \(P\) of \(R_ m\), \(x \in R_ m\) such that \(P \nmid m\) and \(P \nmid x\) let \(\chi_{p} (x)\) the \(m\)-th power residue symbol of \(x\). If \(q = NP = p^ f\) let \(\psi_ p (x)\) be the additive character \(\psi_ p(x) = I_ p^{T_ r(x)}\). If \(n \in \mathbb{Z}\), \((n,m) = 1\), let \(\sigma_ n\) denote the automorphism: \(I_ m \to I^ n_ m\) of \(K_ m/ \mathbb{Q}\). Under the above notation the authors consider the following Jacobi sum \(I_ n(P) = g(P)^ n/g(P)^{\sigma_ n}\), where \(g(f) = \sum_{x \in R_ m/P} \chi_ P (x)\psi_ P(x)\) and prove that if \(n \not \equiv 1 \pmod p\) then \(K_ m = \mathbb{Q} (I_ n(P))\) if and only if \(p \equiv 1 \pmod m\).