A note on Jacobi sums. II (Q1318954)

[Part I, cf. ibid. 69, 32-34 (1993; Zbl 0798.11047).] For an odd prime \(\ell\), let \(k= \mathbb{Q}(\zeta)\) denote the \(\ell\)-th cyclotomic field with \(\zeta= e^{2\pi i/\ell}\). For a prime ideal \({\mathfrak p}\) of \(k\) with \({\mathfrak p} \nmid\ell\), let \(\chi_{\mathfrak p} (x)= ({x\over {\mathfrak p}})_ \ell\) be the \(\ell\)-th power residue symbol in \(k\), and put \(\text{Norm} ({\mathfrak p})= q= p^ f\). So \(fg= \ell-1\). The Jacobi sum (corresponding to the trivial character \((1,1,\dots, 1)\) (\((\ell+1)\)-components)) is defined by \[ J({\mathfrak p}) =-\sum \chi_{\mathfrak p} (x_ 1) \chi_{\mathfrak p} (x_ 2) \cdots \chi_{\mathfrak p} (x_{\ell+1}), \] where the sum runs over all the \((\ell+1)\)-tuples \((x_ 1, x_ 2,\dots, x_{\ell+1})\), \(x_ i\in \mathbb{Z} [\zeta]/ {\mathfrak p}\), such that \(x_ 1+ x_ 2+ \cdots+ x_{\ell+1}= -1\). Let \({\mathcal G}= \text{Gal} (k/\mathbb{Q})\) be the Galois group of \(k\) over \(\mathbb{Q}\). Further, let \({\mathcal G} (J({\mathfrak p})):= \{\sigma\in {\mathcal G}\mid J({\mathfrak p})^ \sigma= J({\mathfrak p})\}\), \({\mathcal G}^* (J({\mathfrak p})):= \{\sigma\in {\mathcal G}\mid (J({\mathfrak p}))^ \sigma= (J({\mathfrak p}))\}\), \(Z({\mathfrak p}):= \{\sigma\in {\mathcal G}\mid {\mathfrak p}^ \sigma= {\mathfrak p}\}\). The result of the paper is formulated in the following theorem. Theorem. (1) If \(f\) is even, then \({\mathcal G} (J({\mathfrak p}))= {\mathcal G}\), i.e., \(J({\mathfrak p})\in \mathbb{Q}\). (2) If \(f\) is odd, then \({\mathcal G}^* (J ({\mathfrak p}))= {\mathcal G} (J({\mathfrak p}))= Z({\mathfrak p})\), so that \(\mathbb{Q} (J({\mathfrak p}))\) is the decomposition field of \({\mathfrak p}\). [For part III, cf. the review below].