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I'm in a bit of a rut. I find myself quite old (20 - old man) and I don't even know high school math properly. And yet I can actually conduct original research (proof: rediscovered a famous theorem in number theory simply by generalizing a totally unseeming olympiad problem and working at it for a few months, until the result popped out as if by magic). However, I spent the past few years solving olympiad problems, so my knowledge stayed at the same level - basic olympiad level (elementary maths and a tiny pinch of calculus). Going to work after high school didn't help. I blame myself, but I'm seeking your help.

Basically, I'm looking to get to the bleeding edge of research in algebraic and analytic number theory and to accumulate the necessary knowledge in the various prerequisite fields of maths. This, in the shortest possible time. Maybe I ask too much of you, but a large chunk of it reduces to finding the right books on abstract algebra and analysis that will help me accumulate vast amounts of knowledge in these fields while starting nearly from scratch. The books need to be challenging (concise) for this to work: I tried reading Apostol's Calculus and found it too straightforward and not "original" enough if that makes sense. I liked Zorich's Analysis much more, which starts from scratch, and concretely, but builds abstracion and sophistication throughout the book, until by the end of volume 2 you have left Rudin in the dust.

Could you recommend similar books for me on Algebra and Number Theory? I would also appreciate some comments on the books (including Zorich's one if possible). Some ideas I have after researching:

Number Theory: Hardy/Wright's theory of numbers; Ireland/Rosen Number Theory

Algebra: Cameron's Intro to Algebra (Starts with rings, which I am told is the "right" way to do things); Kostrikin's book of the same name (starts with groups, but has a similar style to Zorich making it very suitable for my purposes)

Analysis: Only Zorich so far. (Note: I was once told that Zorich is like Spivak and Rudin in one.)

To emphasize: The books must go at a fast pace, and take you a long way. For this to work, the books must be concise. "First course" books (such as Burkill's first course in analysis - recommended to Cambridge first year's in the math tripos) don't work for this. They must be books that contain both a "first course" in the subject as well as the later more challenging material towards the end. I hope this has made it clear enough.

Any replies will be very much appreciated

Thank you very much

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closed as off-topic by user223391, Clarinetist, Charles, Leucippus, Harish Chandra Rajpoot Dec 29 '15 at 4:57

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  • "Seeking personal advice. Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances." – Community, Clarinetist, Leucippus, Harish Chandra Rajpoot
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    $\begingroup$ Consider complementing your strategy of starting quickly from the beginning with time spent going the other way - pick some bleeding edge papers or monographs to read and work backwards through references when you need to understand a tool or definition. $\endgroup$ – Ethan Bolker Dec 29 '15 at 0:53
  • $\begingroup$ Ireland/Rosen's number theory book is one of my favorites. It is pretty introductory, but you still get to do some fairly advanced things by the end. The exercises are very nice and overall the book is well-written. $\endgroup$ – Ben Sheller Dec 29 '15 at 0:55
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    $\begingroup$ Books by Lang, Serre and Rudin are notoriously terse. $\endgroup$ – Gregory Grant Dec 29 '15 at 1:18
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    $\begingroup$ But it's hard to find an algebra book that starts at square one and gets to the cutting edge. Personally I like Hungerford, which will prepare you to read just about anything. $\endgroup$ – Gregory Grant Dec 29 '15 at 1:19
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    $\begingroup$ Don't you find it contradictory looking for "concise" books and offering Zorich as an example? As far as I remember it is almost 15 hundred pages. $\endgroup$ – Artem Dec 29 '15 at 1:59
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I think this is one of the fastest ways possible: Buy Lang's Algebra and Neukirchs Algebraic Number Theory and start reading Neukirchs Algebraic Number Theory and if you don't know a concept well, look up the corresponding chapter in Langs Algebra and read that chapter. If you want to get to cutting edge algebraic number theory I guess you should also learn some algebraic geometry(thats just what I suspect as I am nowhere near cutting edge research algebraic number theory). If you want to do this very fast I would recommend reading Hartshorn's Algebraic Geometry book. To read this it would probably be good to read a book on commutative algebra first for which I would recommend Atiyah- and MacDonald's Introduction to commutative algebra. I guess you can skip learning commutative algebra if you want to be really fast and look up the proofs when needed(but I would not recommend doing this).

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My favorite on Abstract Algebra is Artin's Algebra. It makes very little assumptions about the reader's background and builds everything in clear language from the foundations.

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  • $\begingroup$ Have you read Hungerford? I think it's every bit as good. $\endgroup$ – Gregory Grant Dec 29 '15 at 1:20
  • $\begingroup$ @GregoryGrant must be recent -- I read Artin in school 15 years ago or so $\endgroup$ – gt6989b Dec 29 '15 at 1:28
  • $\begingroup$ @GregoryGrant Hungerford is a graduate book, which explicitly states that an undergraduate course in modern algebra is a prerequisite. $\endgroup$ – Artem Dec 29 '15 at 1:32
  • $\begingroup$ @Artem That's not quite true. Hungerford starts from scratch. Yes most people who read it do so as a second course, but the OP said he wants something that starts from scratch and goes very fast. That's exactly what Hungerford does. In fact it probably starts even more from scratch than any undergraduate book. $\endgroup$ – Gregory Grant Dec 29 '15 at 1:33
  • $\begingroup$ And by the way, I second Artin. If the OP likes Zorich (not the fastest book to digest), in my taste Artin in Algebra is like Zorich in Analysis. $\endgroup$ – Artem Dec 29 '15 at 1:34

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