陳其誠臺灣大學：數學研究所黃月邑Huang, Yueh-YiYueh-YiHuang2007-11-282018-06-282007-11-282018-06-282007http://ntur.lib.ntu.edu.tw//handle/246246/594971950年代，對於數域上的某些擴張，岩澤提出了計算其 class number 的理論。並討論了 cyclotomic 擴張及其他種類的伽羅瓦擴張。在這篇文章中，我將介紹這個定理並給出一些例子。In the 1950's, Kenkichi Iwasawa had constructed his class number formula on extensions of number fields. Iwasawa investigated towers of cyclotomic fields and other kinds of Galois extensions of number fields whose Galois group is isomorphic to the additive group of p-adic integers. In this thesis, I would like to introduce this theory and give some examples.Contents Abstract in Chinese i Abstract ii Contents 1 1 Introduction 2 1.1 Notations and Basic Definitions . . . . . . . . . . . . . . . . . . . . . 3 2 The Iwasawa Algebra 4 2.1 Some Congruence Relations . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 The Map f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 The Map g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 The Isomorphism and The Continuity . . . . . . . . . . . . . . . . . . 8 3 The Structure of -module 11 3.1 Finitely Generated Module . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Nakayama’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 The Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Iwasawa’s Theorem 16 4.1 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Bibliography 22en-US依烏阿沙娃理論Iwasawa Theorythesis