On the class groups of cyclotomic extensions in presence of a solution to Catalan's equation

AbstractCatalan's conjecture states that the equation xp−yq=1 has no other integer solutions but 32−23=1. We investigate the consequences of existence of further solutions (with odd prime exponents p,q) upon the relative class group of the pth cyclotomic extension. We thus obtain several new results which merge into the conditionq≢1mod pandp≢1mod q. This condition is used in the proof of Catalan's conjecture