Algebraic Number Theory, 2/e (Hardcover)

Bringing the material up to date to reflect modern applications, Algebraic Number Theory, Second Edition has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. This edition focuses on integral domains, ideals, and unique factorization in the first chapter; field extensions in the second chapter; and class groups in the third chapter. Applications are now collected in chapter four and at the end of chapter five, where primality testing is highlighted as an application of the Kronecker–Weber theorem. In chapter five, the sections on ideal decomposition in number fields have been more evenly distributed. The final chapter continues to cover reciprocity laws.

New to the Second Edition

• Reorganization of all chapters
• More complete and involved treatment of Galois theory
• A study of binary quadratic forms and a comparison of the ideal and form class groups
• More comprehensive section on Pollard’s cubic factoring algorithm
• More detailed explanations of proofs, with less reliance on exercises, to provide a sound understanding of challenging material

The book includes mini-biographies of notable mathematicians, convenient cross-referencing, a comprehensive index, and numerous exercises. The appendices present an overview of all the concepts used in the main text, an overview of sequences and series, the Greek alphabet with English transliteration, and a table of Latin phrases and their English equivalents.

Suitable for a one-semester course, this accessible, self-contained text offers broad, in-depth coverage of numerous applications. Readers are lead at a measured pace through the topics to enable a clear understanding of the pinnacles of algebraic number theory.

為了反映現代應用，第二版的《代數數論》已經完全重寫和重新組織，以採用新的風格、方法和呈現方式。本版將重點放在第一章的整數環、理想和唯一分解；第二章的域擴展；以及第三章的類群。應用現在集中在第四章和第五章末尾，其中將Kronecker-Weber定理的應用突出為素性測試。在第五章中，關於數域中理想分解的部分已經更均勻地分佈。最後一章繼續涵蓋互反律。

第二版的新內容包括：
- 所有章節的重新組織
- 更完整和深入的 Galois 理論探討
- 對二次二元形式的研究，並比較理想和形式類群
- 更全面的 Pollard 立方因數分解算法部分
- 更詳細的證明解釋，較少依賴練習，以提供對具有挑戰性的材料的深入理解

本書包括著名數學家的小傳、方便的交叉引用、全面的索引和眾多練習題。附錄提供了主要內容中使用的所有概念概述、序列和級數概述、希臘字母及其英文音譯、以及拉丁短語及其英文對應的表格。

適合一學期的課程，這本易於理解、自成一體的教材提供了廣泛而深入的多個應用範疇的涵蓋。讀者以適度的速度引導通過各個主題，以便清晰理解代數數論的頂峰。