The orders of solutions of the Kummer system of congruences

Abstract

A new method concerning solutions of the Kummer system of congruences (K) (modulo an odd prime l) is developed. This method is based on the notion of the Stickelberger ideal. By means of this method a new proof of Pollaczek’s and Morishima’s assertion on solutions of (K) of orders 3, 6 and 4 mod l \bmod \; l is given. It is also shown that in case there is a solution of ( K ) ≢ 0 , ± 1 ( mod l ) (K)\, \not \equiv \, 0, \pm 1\;\pmod l , then for the index of irregularity i ( l ) i(l) of the prime l we have i ( l ) ≥ [ l / 2 3 ] i(l) \geq [\sqrt [3]{{l/2}}] .