
Advanced Algebra textbooks - efm
http://www.math.stonybrook.edu/~aknapp/download.html
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eliteraspberrie
I 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.

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g9yuayon
Can'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.

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tnecniv
I 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.

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craigching
I'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)

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danharaj
The 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)

~~~
navi54
Do you have any suggestions on other topics and books in math? This books is
amazing.

~~~
danharaj
Math is too big. Got a particular desire?

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lumberjack
Eh, 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.

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gaur
Book 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".

~~~
danharaj
I 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.

~~~
gaur
> they all have reasonable need of an introduction to abstract algebra.

Then the word "abstract" should appear in the title.

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danharaj
Well, can't argue with that.

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mdergosits
Somewhat 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

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eccstartup
I 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
FYI,
[http://www.genealogy.ams.org/id.php?id=8127](http://www.genealogy.ams.org/id.php?id=8127)

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catnaroek
I'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.

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gaur
Looks 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
Can you elaborate on that? I find computer modern much more pleasant to read
than Times.

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gaur
It'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.

