An easier proof of the Mazur-Wiles theorem can be found by using Euler systems as developed by Kolyvagin ( see Washington 's book ) . Other generalizations of the main conjecture proved using the Euler system method have been found by Karl Rubin , amongst others . Karl Rubin found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin 's Euler systems , described in and , and later proved other generalizations of the main conjecture for imaginary quadratic fileds .