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Advanced Algebra textbooks
- lumberjack 11y agoEh, is this supposed to be a good book because I have no idea why the definitions aren't clearly marked and indexed. I only checked the chapter about Group Theory but I was not impressed. Maybe for a quick review of the topic it might be enough but for a beginner it seems that it is not rigorous. The definitions could be much more clear explicit. And there is no reason why they should not be indexed.
- catnaroek 11y agoI'm by no means a typography nerd, but the font used in both PDFs is really bad. Not only is it visually unpleasant; most importantly, it is genuinely hard to read.
- gaur 11y agoLooks like Times to me. It's a standard mathematical font, and anyway is more pleasant and easier to read than the abominable Computer Modern fonts.
- andrepd 11y agoCan you elaborate on that? I find computer modern much more pleasant to read than Times.
- gaur 11y agoIt's an inferior clone of Monotype Modern, just like Arial is an inferior clone of Helvetica. In terms of being easier to read, I find the extreme contrast between thick and thin strokes does not work well on a screen.
- eliteraspberrie 11y agoI don't like the typical definition-theorem-proof approach of most textbook in mathematics, including these. It's great for a classroom, no good for self-study. As an alternative, I highly recommend A Book of Abstract Algebra by Pinter. If you work through that first, you may actually enjoy these two later.
- codeofthedamned 11y agoI agree. Pinter's book is very easy to read and great for self-study.
- tzs 11y agoAnd it's a Dover book, so is quite reasonably priced (around $12).
- g9yuayon 11y agoCan't agree more. Definition-theorem-proof type of textbooks is way too clean. They don't tell you how ideas came to be or why they mattered. In other words, it's hard for students to learn the intuitions behind the ideas. I wish there are list of "XXX from Ground-Up" type of books that show readers a list of problems, struggles of people trying to solve them, and how ideas emerge from the numerous attempts. Leslie's paper Paxos Made Simple was written in that way. A few chapters of Kleinberg's Algorithm Design were written in that way too.
- tnecniv 11y agoI think math classes should be paired with history more. My probability professor often offered historical context (for example, the Poisson distribution first being used to model deaths due to horse kicks in the Prussian army) to the ideas we discussed, and the stories were often both interesting and insightful.
- jostylr 11y agoNature and Growth of Mathematics by Edna Kramer is a fantastic book interweaving history and math.
- craigching 11y agoI've been refreshing on Calculus and I found that Kline's book was good at application as well as a bit of history, at least I never got the history part at University and I found it very interesting. http://store.doverpublications.com/0486404536.html http://store.doverpublications.com/0486404536.html
- eccstartup 11y agoI thought it was only a fancy book. But it turns out to be a serious one. I will read this book, especially the advanced one.
- eccstartup 11y agoFYI, http://www.genealogy.ams.org/id.php?id=8127 http://www.genealogy.ams.org/id.php?id=8127
- danharaj 11y agoThe best undergrad algebra textbook I've studied is Algebra by Mac Lane and Birkhoff (3rd edition! the previous editions aren't quite as good and substantially different; i haven't seen the 4th edition and it is out of print so /shrug). I've used multiple books both in self-study and class and this book is, to me, in a league of its own. Not only does Algebra teach modern algebra, it teaches one to think like a modern algebraist, and not like just any modern algebraist, but like Saunders Mac Lane who was pretty great at algebra. As an example of Algebra's approach, take the isomorphism theorems [1]. Now many undergraduate textbooks (like Dummit and Foote) will prove these theorems by manipulating cosets and deal with gross "implementation details" at the level of sets. Mac Lane insists otherwise: The only time you have to manipulate cosets is in order to construct the quotient G/N of a group G by one of its normal subgroups N. Once you have constructed this group and proved its universal property, the isomorphism theorems can be proved without ever mentioning cosets again. What is that universal property? It has two parts: First is that there is a morphism p from G to G/N which sends all of N to the identity in G/N. Second is that any morphism f from G to any group L that sends all of N to the identity in L necessarily factors uniquely up to isomorphism as a composition of morphisms g ∘ p. This is the essence of a quotient group. Mac Lane's approach is to apprehend the essence of what is studied while discarding as much of the set theoretic husk as is possible. It is algebra in its purest form, accessible to and transformative of the mind of an undergraduate. Reading this book is a recurring joy to me. [1] https://en.wikipedia.org/wiki/Isomorphism_theorem https://en.wikipedia.org/wiki/Isomorphism_theorem
- birdperson 11y agoI second this. This a phenomenal book. I didnt like it at first sight because I thought it looked kind of ugly, but the authors have a knack for clarifying some of the more arcane concepts in math in a very few words. It's not a cutesy bestseller that will end up teaching you jackshit, nor it's an intimidating monster like Lang's Algebra that's meant to put hair on your chest. This book is just right.
- greydius 11y agoI also found this book to be excellent.
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- gaur 11y agoBook titles like these are just more evidence that some mathematicians don't understand (or willfully misconstrue) the meaning of words like "basic" or "introduction". "Basic Algebra" means "material typically covered in late middle or early high school".
- analog31 11y agoOn an amusing note, I noticed the title, and decided to show my daughter, who is in high school. She knew that I was joking. I had already asked her, after her first day of algebra class, if she had learned about groups.
- danharaj 11y agoI disagree. The word "introduction" immediately introduces ambiguity: Introduction to whom? An introduction to the aspiring physicist will certainly look drastically different from one to a high school student or an undergraduate mathematics student, but they all have reasonable need of an introduction to abstract algebra.
- eastWestMath 11y agoI feel like mathematicians are the ones who get to decide what constitutes basic algebra.
- eastWestMath 11y agoI feel like mathematicians are the ones who get to decide what constitutes basic algebra.
- mdergosits 11y agoSomewhat unrelated, but I enjoy the logical dependence chart. It's nice to see the ordering of the topics. Rather than trying to figure that out for oneself