Browsing by Author "Ikonen, Sakari"

This thesis covers the factorization properties of number fields, and presents the structures necessary for understanding a proof on Iwasawa's theorem. The first three chapters aim to construct a ring of integers for arbitrary number fields, and prove that such a ring exists. We prove that our ring of integers is a Dedekind ring, giving us unique factorization on the set of prime ideals. We prove that there exists an isomorphism between principal and factorial divisors and ideals, define an equivalence relation on the set of all divisors, and show that the equivalence classes form the ideal class group. The class number of a field is defined as the order of the ideal class group. We define ramification of primes, and the invariants related to a prime P called the ramification index, inertia degree and decomposition number. We expand on the Galois theory of finite extensions, by introducing a topology on an infinite algebraic Galois extension, and a Galois correspondence between closed subgroups and intermediate fields. We show how to define the decomposition- and inertia group in the infinite case. The maximal unramified field extension, the Hilbert class field, whose Galois group is isomorphic to the ideal class group, is introduced. We introduce a p-adic metric on the ring of integers with the help of valuations, and construct the p-adic integers as a completion with regards to the metric. We prove some structure results for this ring. The lambda-modules are constructed as a limit of modules over group rings, where the group rings are generated by the p-adic integers, and a suitable multiplicative cyclic group. The final result is a proof of Iwasawa’s theorem as found in Washington, Introduction to Cyclotomic fields. We view the Galois group of the p-adic extension as a lambda-module, and from the structure theorems of lambda-modules, we prove results that carry on to the galois groups of the intermediate fields, culminating in a formula for the exact power of p, that divides the class number of the n-th intermediate field.