ETD

• Iwasawa Theory
• Number Theory
• Zp-extensions

In this dissertation, we explain some of the classical results concerning the theory of Zp-extensions of a number fields, that was started by K. Iwasawa in the 1950s. We begin by explaining the theory of modules over the Iwasawa Algebra and use it to derive Iwasawa's asymptotic formula for the p-part of the class number of the intermediate fields in a Zp-extension in terms of the so called λ- and μ-invariants of the extension. We then move to explain a result by Ferrero and Washington stating that μ = 0 for the cyclotomic Zp-extension of an abelian number field; this requires to construct p-adic analogues of the classical complex L-functions associated with Dirichlet characters; we do so following Mazur’s approach via p-adic measures, and use the language of p-adic measures to explain Sinnott’s reduction of Ferrero-Washington theorem to a relation between the μ-invariant of a p-adic measure and the one of its Leopoldt’s Gamma transform. Throughout our dissertation, we explore multiple examples of computations for λ- and μ-invariants of number fields.