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I'm interested in a solid list of the greatest textbooks in the major* fields as written by the "Masters".

The problem I have is that after many years of a high-level, rigorous education in mathematics in I have encountered the same problem many aspiring mathematicians before me have, namely the realization that much of our knowledge of mathematics we have learned to this point is on pretty shaky ground. I understand that the more expository textbooks are perhaps "necessary" to get enough mathematical maturity to grasp the motivations for the subject and later on understanding that these textbooks are too elementary and incomplete and a more rigorous treatment is needed, but now that I have gotten to that point I have the need to read the books that put the mathematics that I have learned in as close to foundational perfection as has been achieved in each field.

But I have had a difficult time deciding which books to read for the reason illustrated in the following example: While Euclid is one of the greatest masters in history and I have read part of the Elements, Foundations of Geometry by Hilbert is much better in the sense that it greatly polishes (not dumbs down or make easier) a lot of the hand-waving of Euclid and sets Geometry in a much firmer logical ground. Therefore, in my sense, reading Foundations would be more useful to my goals than reading Elements. The point being that the book should be the most logically complete and close to the highest accomplishment in setting its respective field in solid, rigorous ground. (I emphasize that I'm interested in the best textbooks regardless of level of difficulty or terseness. For example, Baby Rudin and higher levels of difficulty are fine.)

  • By major I mean: Euclidean Geometry, Higher geometries, Algebra, Analysis, Number Theory, and the like.

Thanks.

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closed as too broad by Eric Wofsey, Antonio Vargas, wythagoras, TravisJ, user99914 Sep 14 '15 at 21:06

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ A paragraph break here and there can make your post more readable. $\endgroup$ – Thomas Andrews Sep 13 '15 at 23:48
  • $\begingroup$ Perusing the site for specific subjects will reveal lists of some of the most-highly regarded texts for each subject. Why not compile a list of the "greatest textbooks" that way? $\endgroup$ – Clayton Sep 14 '15 at 0:00
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EGA is pretty self contained and complete. Also it has a good exposition. Bourbaki's Elements of Mathematics also satisfy your requirements, especially the amazing Lie groups and Lie algebras.

Serre's books are mostly not self contained, but his Course in Arithmetic is fairly accessible with an excellent selection of topics.

Another user already mentioned Milnor's books, which are splendid and understandable by humble mortals like me.

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Try "Characteristic Classes" of John W. Milnor and James D. Stasheff, which is easy to spot.

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