Abstract
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 .