I fell on the following fact : let p be an odd prime, let K denote the p-th cyclotomic field, let L be an extension of K with finite degree not divisible by p, and assume that the prime ideal $(1 - \zeta)$ of K (where $\zeta$ denotes a primitive p-th root of unity) ramifies completely in L. Let P denote the only prime ideal of L dividing $(1 - \zeta)$. Then in every solution (if any) of the equation $x^p + y^p = z^p$ where x, y and z are P-integral elements of L not divisible by P, the rational integer t congruent to x/y modulo P is a root of the Kummer-Mirimanoff system of congruences $B_{2i}l^{p-2i}(t + \zeta) \equiv 0$ (mod p) for i = 1 to (p-1)/2. Do you know if this was already published ? Thanks in advance.