Fermat's equation over the tower of cyclotomic fields

Let be a prime, let let be the maximal real subfield of , and let be the maximal -subextension of . We define effectively calculable integer-valued functions , and such that , where is the index of irregularity of . For we prove the first case of Fermat's theorem for , , and . We obtain explicit lower estimates for , and . For regular (when ) we prove the second case of Fermat's theorem for and and Fermat's theorem for , and , generalizing the classical result on the validity of Fermat's theorem for and regular . We also obtain some other results on solutions of Fermat's equation over , and .