Aluffi P. Algebra. Notes From The Underground 2021

Textbook in PDF format

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.
Treats all standard topics for a first course in detail, moving from rings to modules to groups to fields, which allows for a more gradual entry into the subject
Emphasizes a modern perspective, with hints toward category theory, exposing students to advanced topics and preparing them for graduate courses and applications of algebra in other subjects
Includes 400+ exercises, with 150 worked-out problems from the main text, and plenty of examples to help students better understand the material
Uses a conversational style and strong narrative flow that makes the material easier for students to follow and appreciate as a whole