Abstract Algebra: A First Course

When a student of mathematics studies abstract algebra, he or she inevitably faces questions in the vein of, "What is abstract algebra" or "What makes it abstract?"

Algebra, in its broadest sense, describes a way of thinking about classes of sets equipped with binary operations. In high school algebra, a student explores properties of operations (+, -, ×, and ) on real numbers.

Abstract algebra studies properties of operations without specifying what types of number or object we work with. Any theorem established in the abstract context holds not only for real numbers but for every possible algebraic structure that has operations with the stated properties.

This textbook intends to serve as a first course in abstract algebra. The selection of topics serves both of the common trends in such a course: a balanced introduction to groups, rings, and fields; or a course that primarily emphasizes group theory.

The writing style is student-centered, conscientiously motivating definitions and offering many illustrative examples. Various sections or sometimes just examples or exercises introduce applications to geometry, number theory, cryptography and many other areas.

This book offers a unique feature in the lists of projects at the end of each section. the author does not view projects as just something extra or cute, but rather an opportunity for a student to work on and demonstrate their potential for open-ended investigation.

The projects ideas come in two flavors: investigative or expository. The investigative projects briefly present a topic and posed open-ended questions that invite the student to explore the topic, asking and to trying to answer their own questions.

Expository projects invite the student to explore a topic with algebraic content or pertain to a particular mathematician's work through responsible research.

The exercises challenge the student to prove new results using the theorems presented in the text. The student then becomes an active participant in the development of the field.

當數學學生學習抽象代數時，他們不可避免地會面臨類似「什麼是抽象代數」或「什麼使它成為抽象」的問題。

代數在最廣義的意義上描述了一種思考配備二元運算的集合類別的方式。在高中代數中，學生探索實數上的運算（+、-、×和÷）的性質。

抽象代數研究的是不指定我們使用的數字或物件類型的運算性質。在抽象上下文中建立的任何定理不僅適用於實數，而且適用於具有所述性質的每種可能的代數結構。

本教科書旨在作為抽象代數的第一門課程。所選的主題既符合這門課程中的兩種常見趨勢：平衡介紹群、環和域；或者主要強調群論的課程。

寫作風格以學生為中心，謹慎地解釋定義並提供許多例子。各個章節或有時僅是例子或練習引入了對幾何、數論、密碼學和許多其他領域的應用。

本書在每個章節結尾提供了一個獨特的特點，即項目列表。作者不僅將項目視為額外或可愛的東西，而是將其視為學生進行開放式調查並展示他們潛力的機會。

項目的想法有兩種風格：調查性或闡述性。調查性項目簡要介紹一個主題並提出開放式問題，邀請學生探索該主題，提出並嘗試回答自己的問題。

闡述性項目邀請學生通過負責任的研究探索具有代數內容或與特定數學家的工作相關的主題。

練習挑戰學生使用教科書中提出的定理證明新結果。學生因此成為該領域發展的積極參與者。

Stephen Lovett is an associate professor of mathematics at Wheaton College. He earned a PhD from Northeastern University. He is a member of the Mathematical Association of America, American Mathematical Society, and Association of Christians in the Mathematical Sciences.. His research interests include commutative algebra, algebraic geometry, differential geometry, cryptography, and discrete dynamical systems. Dr. Lovett's other books are: Differential Geometry of Curves and Surfaces, Third Edition with Thomas F. Banchoff, Differential Geometry of Manifolds, Second Edition, and Transition to Advanced Mathematics, with Danilo R. Diedrichs, all from CRC Press.

Stephen Lovett 是 Wheaton College 的副教授，專攻數學。他在 Northeastern University 取得博士學位。他是美國數學協會、美國數學學會和基督徒數學科學協會的成員。他的研究興趣包括交換代數、代數幾何、微分幾何、密碼學和離散動力系統。Lovett 博士的其他著作包括與 Thomas F. Banchoff 合著的《曲線和曲面的微分幾何，第三版》，與 Danilo R. Diedrichs 合著的《流形的微分幾何，第二版》，以及《進階數學導論》（與 Danilo R. Diedrichs 合著），這些都是由 CRC Press 出版的。