Algebra: Notes from the Underground

This text provides an undergraduate introduction to abstract algebra, emphasizing a modern perspective and covering topics such as modules, Noetherian modules, exact complexes, and standard results on field extensions. It features a conversational style, 400+ exercises, and an appendix with complete solutions.

Format: Paperback / softback
Length: 488 pages
Publication date: 03 June 2021
Publisher: Cambridge University Press

Abstract algebra is a fascinating subject that encompasses a wide range of topics, from rings to modules to groups to fields. This undergraduate introduction to abstract algebra takes a unique approach, emphasizing a modern perspective on the subject while also touching on the unifying categorical principles that underpin various constructions.

One of the key features of the text is its treatment of modules. It includes a proof of the classification theorem for finitely generated modules over Euclidean domains, which is a fundamental result in the field. Additionally, Noetherian modules and some of the language of exact complexes are introduced, providing students with a solid foundation for further study.

In addition to covering 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, the text also delves into more advanced concepts. It discusses the theory of homological algebra, which is a powerful tool for studying modules and rings, and it introduces the concept of sheaves, which are a generalization of modules that are useful in various areas of mathematics.

The text is written in a conversational style, making it accessible to students with a variety of backgrounds. It includes over 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. This additional material helps students to deepen their understanding of the subject and to apply the concepts they have learned in real-world situations.

Overall, this undergraduate introduction to abstract algebra is an excellent resource for students who are interested in learning about this fascinating subject. It provides a solid foundation for further study and offers a modern perspective on the subject while also touching on the underlying principles that unite the various constructions. Whether you are a student of mathematics or simply someone who enjoys exploring the world of abstract algebra, this text is sure to be a valuable addition to your library.

Weight: 838g
Dimension: 171 x 244 x 28 (mm)
ISBN-13: 9781108958233