Class group behaviour in cyclotomic extensions of abelian fields

Let K be a number field. Understanding the ideal class group Cl_K of the field K is a classical problem in algebraic number theory. Solving this problem is hard, especially if the discriminant of the field K is large. However, we can get interesting results focusing on the q-part of Cl_K for a fixed prime q rather than focusing on the full class group. In this thesis we restrict our attention to the fields appearing in cyclotomic Z_p-extensions of abelian number fields K. In the first part we study the Greenberg's conjecture for a real quadratic field and its cyclotomic Z_2-extension. In particular, we present an algorithm to check if the conjecture is true for the cyclotomic Z_2-extensions of the fields F = Q(sqrt(f)) where f is a positive integer and f < 10000. In the second part of the thesis we study the q-part of the class number in Z_p-extensions with q is different from the prime p and q is odd.