Algebra

Notes from the Underground

Paolo Aluffi

• Publisher: Cambridge University Press
• Year: 2021
• Print ISBN: 9781108958233
• eBook ISBN: 9781108962698
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

From rings to modules to groups to fields, this undergraduate introduction to abstract algebra follows an unconventional path. The text emphasizes a modern perspective on the subject, with gentle mentions of the unifying categorical principles underlying the various constructions and the role of universal properties. A key feature is the treatment of modules, including a proof of the classification theorem for finitely generated modules over Euclidean domains. Noetherian modules and some of the language of exact complexes are introduced. In addition, standard topics - such as the Chinese Remainder Theorem, the Gauss Lemma, the Sylow Theorems, simplicity of alternating groups, standard results on field extensions, and the Fundamental Theorem of Galois Theory - are all treated in detail. Students will appreciate the text's conversational style, 400+ exercises, an appendix with complete solutions to around 150 of the main text problems, and an appendix with general background on basic logic and naïve set theory.

Table of contents

• Cover
• Half-title
• Series information
• Title page
• Copyright information
• Contents
• Introduction
• Before we Begin
• Part I Rings
• Part II Modules
• Part III Groups
• Part IV Fields
• Appendix A Background
• Appendix B Solutions to Selected Exercises
• Index of Definitions
• Index of Theorems
• Subject Index