On the class groups of cyclotomic extensions in presence of a solution to Catalanʹs equation Original Research Article

Catalanʹs 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 conditionimage This condition is used in the proof of Catalanʹs conjecture.