Algebraic Number Theory for Beginners Algebraic Number Theory for Beginners: Following a Path from Euclid to Noether

This book introduces algebraic number theory through the problem of generalizing 'unique prime factorization' from ordinary integers to more general domains. Solving polynomial equations in integers leads naturally to these domains, but unique prime factorization may be lost in the process. To restore it, we need Dedekind's concept of ideals. However, one still needs the supporting concepts of algebraic number field and algebraic integer, and the supporting theory of rings, vector spaces, and modules. It was left to Emmy Noether to encapsulate the properties of rings that make unique prime factorization possible, in what we now call Dedekind rings. The book develops the theory of these concepts, following their history, motivating each conceptual step by pointing to its origins, and focusing on the goal of unique prime factorization with a minimum of distraction or prerequisites. This makes a self-contained easy-to-read book, short enough for a one-semester course.

這本書通過將「唯一質因數分解」從普通整數推廣到更一般的領域，介紹了代數數論。在整數中解多項式方程自然地導致了這些領域，但在這個過程中可能會丟失唯一質因數分解。為了恢復它，我們需要戴德金的理想概念。然而，仍然需要代數數域和代數整數的支持概念，以及關於環、向量空間和模的支持理論。埃米·諾特（Emmy Noether）將使唯一質因數分解成為可能的環的性質總結在一起，我們現在稱之為戴德金環。本書按照這些概念的發展歷史，通過指出每個概念步驟的起源來激發讀者的興趣，並專注於以最少的干擾或先備知識實現唯一質因數分解的目標。這使得本書成為一本自成一體、易於閱讀的書籍，篇幅足夠短，適合一學期的課程使用。