On the relation between units and Jacobi sums in prime cyclotomic fields (Q1191492)

Let \(p\geq 5\) be a prime number, \(\zeta_p\) a primitive \(p\)-th root of unity. Let \[ \beta_ r=\prod^{p-1}_{k=1}((1-\zeta^ k_ p)/(1- \zeta_ p))^{k^{(p-1-r)p^{n-1}}} \] for an even integer \(r\) with \(2\leq r\leq p-3\). Let \(Q\) be a prime ideal of \(\mathbb{Z}[\zeta_p]\) prime to \(p\). Let \(n\geq 1\) be an integer such that \(p^n\) divides \(|(\mathbb{Z}[\zeta_p]/Q)^ \times|\). In the present article the author studies the indices \(i(Q)\) of units \(\beta_r\) with respect to \(Q\). First in Theorem 1, he gives an expression for \(i(Q)\pmod{p^n}\) in terms of the logarithmic derivatives of certain Jacobi sums, which is an extension of a formula stated by Kummer in 1852 relating to Fermat's last theorem. Next in Theorem 2, he calculates the logarithmic derivatives of Jacobi sums appearing in this expression. To do this the indices \(i(\mathbb{B})\) of prime ideals \(\mathbb{B}\) of \(\mathbb{Z}[\zeta_{p^n}]\) are studied. From these results several consequences about relations of \(i(\mathbb{B}_k)\) for prime ideals \(\mathbb{B}_k\) appearing in the prime ideal factorization of some principal ideals in \(\mathbb{Z}[\zeta_{p^n}]\) are proved. For example, an expression of \(\sum^\ell_{k=1} a_ki(\mathbb{B})\), where \(\prod^\ell_{k=1}\mathbb{B}_k^{a_k}\) is the prime ideal factorization of \((\delta^{p^{n- 1}}+\varepsilon^{p^{n-1}}\zeta_{p^n})\) in \(\mathbb{Z}[\zeta_{p^n}]\) for some \(\delta\) and \(\varepsilon\), is obtained, which is considered as an extension of Kummer-Mirimanoff congruences for the first case of Fermat's last theorem.