• triviality
• non-p-part
• Iwasawa
• cyclotomic
• class number
• circular units
• Z_p-extension

Inside this thesis one can find a study, based on the work of professor Kuniaki Horie, of the non-p-part of the class number of a cyclotomic Z_p-extension over Q or an imaginary quadratic field. Starting from the well-known theorem of Washington which states that for this kind of fields the l part of the class number (for l a prime different from p) is bounded, one wonders when it is actually trivial. By rewriting the class number in terms of the index of the group of circular units and applying the analytic class number formula for CM-fields, we prove that triviality occurs eventually outside some arithmetic progression. An effective bound depending on the length of the progression is also provided.