An Invitation to Abstract Algebra

Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics.

To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout.

The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois' achievement in understanding how we can--and cannot--represent the roots of polynomials.

The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory.

The presentation includes the following features:

• Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters.
• The text can be used for a one, two, or three-term course.
• Each new topic is motivated with a question.
• A collection of projects appears in Chapter 23.

Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks--period. This book is offered as a manual to a new way of thinking. The author's aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.

學習抽象代數可以是一個令人驚嘆的探索冒險。這個主題不需要被淡化，也不應該被呈現得好像所有的學生都會成為數學教師一樣。這是一個美麗、深奧且實用的領域，是數學內外許多領域共同的語言一部分。

要開始這個探索之旅，對數學推理有一些經驗是有益的。本書對其主題採取了相當嚴謹的方法，並期望讀者能夠理解和創造證明，以及在整個過程中舉出例子。

這本書遵循一個單一的軌跡，從算術和高中代數的謙卑開始，逐漸引入抽象結構和概念，最終達到尼爾斯·亨利克·阿貝爾和埃瓦里斯特·伽羅瓦在理解如何以及不能表示多項式的根方面的成就。

有數學經驗的讀者可能會注意到對於交換代數和對數的喜愛。

本書的介紹包括以下特點:
- 練習設計旨在支持和擴展章節中的材料，以及為後續章節做準備。
- 本書可用於一學期、兩學期或三學期的課程。
- 每個新主題都以一個問題作為動機。
- 第23章包含一系列的專案。
- 抽象代數確實是一個深奧的主題；它不僅可以改變人們對數學的思考方式，還可以改變人們的思考方式。本書旨在成為一本引導新思維的手冊。作者的目標是激發對材料的理解的渴望，鼓勵更多的發現，並培養對主題本身的欣賞。

Steve Rosenberg is a professor in the Mathematics and Computer Science Department at the University of Wisconsin-Superior. He received his Ph.D. from the Ohio State University. As an educator, Dr. Rosenberg has both developed and taught a wide array of courses in mathematics and computer science. As a researcher, he has published results in the areas of algebraic number theory, cryptographic protocols, and combinatorial designs, among others. As a software developer, his clients included Coca-Cola Enterprises and the pension agency of Cook County Illinois. He has extensive experience in computer science and software engineering.

Steve Rosenberg是威斯康辛大學超級理工學院數學與計算機科學系的教授。他在俄亥俄州立大學獲得博士學位。作為一名教育工作者，Rosenberg博士開發並教授了多門數學和計算機科學課程。作為一名研究人員，他在代數數論、加密協議和組合設計等領域發表了研究成果。作為一名軟體開發者，他的客戶包括可口可樂企業和伊利諾伊州庫克縣的退休金機構。他在計算機科學和軟體工程方面擁有豐富的經驗。