Algebra: Notes from the Underground

Algebra: Notes from the Underground

  • Downloads:9726
  • Type:Epub+TxT+PDF+Mobi
  • Create Date:2021-07-30 09:51:07
  • Update Date:2025-09-07
  • Status:finish
  • Author:Paolo Aluffi
  • ISBN:1108958230
  • Environment:PC/Android/iPhone/iPad/Kindle

Summary

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 na�ve set theory。

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Reviews

Iason Szatan

This probably won't replace the encyclopedic Dummit&Foote or the ever-popular Gallian, but it can definitely replace the ageing Fraleigh, and will possibly replace Pinter and/or Artin。 This probably won't replace the encyclopedic Dummit&Foote or the ever-popular Gallian, but it can definitely replace the ageing Fraleigh, and will possibly replace Pinter and/or Artin。 。。。more

henry

Reviewed Wednesday 19 May 2021; I was provided with an advance preorder copy。The author of the wildly successful 'Algebra: Chapter 0', a beginning graduate algebra textbook with a distinctive category-theoretic flavour, has now produced an undergraduate textbook of equal or greater quality, now with Cambridge University Press。 The textbook has the familiar style of lecture notes from an experienced instructor。 What might have been the distinctive feature of the previous book, the mainstreaming o Reviewed Wednesday 19 May 2021; I was provided with an advance preorder copy。The author of the wildly successful 'Algebra: Chapter 0', a beginning graduate algebra textbook with a distinctive category-theoretic flavour, has now produced an undergraduate textbook of equal or greater quality, now with Cambridge University Press。 The textbook has the familiar style of lecture notes from an experienced instructor。 What might have been the distinctive feature of the previous book, the mainstreaming of category-theoretic language into the text, is not really present here (indeed, a definition of a category is absent from the text by design), but categories are 'latent' in the background。 As the author puts it, it's not that category theory was 'put in' to the other book, it's just hard nowadays to take it out。 The 'rings first' approach to teaching abstract algebra is unsurprising here, as the author takes pains to introduce concepts in increasing levels of abstraction, only when necessary, and always with clarity of explanation in mind。 I personally find that his rings-modules-groups-fields sequencing to be rather convincing。 Indeed, should the author be as successful as he deserves to be, then he might yet establish a trend towards 'rings first' being the standard approach in undergraduate courses (if it is not already)。 With the proliferation of introductory 'groups and symmetry' books (e。g。 Armstrong, Carter) it seems that at least one alternative to teaching 'groups first' in abstract algebra is to cover groups and symmetry in a separate course suitable for the first year of undergraduate study, as is now standard at many universities。 Exercises seem to be of high quality, although with a new book the expectation is that things might need to be switched in and out and moved around as the book is used successfully for teaching real students in real mathematics courses。 The fact that this book has already been road-tested on students, as it derives from previously extant lecture notes, is a plus。 Overall, the availability of a new abstract algebra book of extremely high quality seems like a wonderful achievement。 I would recommend that instructors seriously consider using this book for abstract algebra courses in the second or third year of undergraduate study, and the author's other 'Algebra: Chapter 0' for courses in the third or fourth year of undergraduate study, as appropriate。 。。。more