Let \(\ell\) be an odd prime. The author considers the solutions of the following Kummer’s system of congruences: \[ \phi_{\ell -2j}(t) B_{2j} \equiv 0 \pmod {\ell},\quad j=1,\ldots,(\ell -3)/2, \] where \(\phi_ i(t)\) denote the Mirimanoff polynomials and \(B_{2j}\) the Bernoulli numbers. He introduces another system of congruences having essentially the same solutions. This system depends on the Stickelberger ideal of the group ring \({\mathbb Z}[G]\), where \(G\) is the cyclic group of order \(\ell -1\). The solutions are then shown to satisfy the further congruences \[ \sum^{N-1}_{u=1}t^ u\sum^{Y}_{x=1}x^{\ell - 2} \equiv 0\pmod \ell, \] where \(N\) is any integer in the interval \(2,\cdots,\ell -2\) and \(Y\) is an integer depending on \(u\) and \(N\) (in an explicitly given but complicated way). For \(N=2\) and \(N=3\) this yields Wieferich’s and Mirimanoff’s criteria for the first case of Fermat’s last theorem.