Algebra : Notes from the Underground PDF

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 naive set theory.