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I acknowledge my limited knowledge of abstract algebra(My background comprising groups and subgroups from Herstein's Topics in Algebra is hardly worth mentioning) .And yet, I confess I really like numbers and primes.In short, I wish to explore algebraic number theory which I am told needs a definite background.I will be happy to know what exactly I need to know before I can study algebraic number theory.

Edit:I am not seeking any shortcuts.I am talking about studying something really concrete.

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Basically, commutative algebra and some field theory. For a first course, you should know about rings/fields, integral extensions, Galois groups, Galois extensions, finitely generated Z-modules, etc. –  Fredrik Meyer Jul 29 '12 at 18:06
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Galois theory for sure; know it with proofs really really well, you will use it constantly. Commutative algebra, although you can probably just believe some foundational results the first time through. Linear algebra over $\mathbb{Z}$, which is harder than it should be the first 200 times you see it. –  user29743 Jul 29 '12 at 18:13
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Since algebraic number fields are relatively concrete objects in mathematics, in principle, you don't need much knowledge about abstract algebra. For example, Dirichlet-Dedekind's lectures on the theory of numbers presupposes no abstract algebra because there was no such thing as abstract algebra at that time. –  Makoto Kato Jul 29 '12 at 18:15
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I think modern algebraic numbers theory requires a hefty knowledge of abstract algebra. That the Dirichlet-Dedekind don't use it says nothing, as what they do is hardly modern alg. number theory, just like the Elements by Euclides is hardly modern geometry. It'd be pretty hard to get into business without Galois Theory -- including, of course, group and ring theory, modules, linear algebra and etc. –  DonAntonio Jul 29 '12 at 19:00

3 Answers 3

up vote 5 down vote accepted

If you're willing to put in the work, I recommend the following sort of three-step plan: [1] get Dummit and Foote's Abstract Algebra and [2] get Ireland and Rosen's Classical Introduction to Modern Number Theory. Ireland and Rosen develops a very nice look towards number theory from a largely algebraic viewpoint, and it's pretty gentle.

Here's the trick: I don't really know how much abstract algebra you know, but Dummit and Foote is miraculously good at serving as an early reference. If you want to know about, say, fields, you can open up the fields chapter and start reading, and you'll be fine. It's not notationally cumbersome or overly self-referential. (I'm trying to distinguish this from books like Folland's Real Analysis: no proof is more than paragraph, almost, because every proof looks like "Use theorem 16.4(a) to show such and such, then argue as in the proof of 3.2.1 to do so and so, and conclude similarly to the end of corollary 4.2", which is great if you know those things and terrible if you don't. Side note - I do like Folland).

So you can go through Ireland and Rosen, referring to Dummit and Foote when you need. I would encourage you to learn some of the module/ring/field/galois theory while you learn some algebraic number theory, as these will become necessary to understand algebraic number theory eventually and are harder to simply refer to for reference.

Finally, since I'm sure you've been waiting for the third step of the three-step-plan, I'd recommend finding a professor/mentor/student to work with, so that when you come across something you haven't seen before, you can ask where to look. I see that Ragib wrote a similar-in-feel answer while I wrote my answer up, and there's a certain similarity in our recommendation: you can probably start now, but learn some algebra while you progress.

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I have started ring theory from Dummit and Foote's book and I dare say,it is clear and amazing.Thanks for the reference. –  user31029 Sep 17 '12 at 8:17

Many comments above seem too pessimistic. You will need it eventually, but you don't need to start off as a master of Commutative algebra and Galois theory to start enjoying algebraic number theory. I enjoy the text "Algebraic Number Theory" by Richard Mollin which is ideal for self study (lots of examples and exercises) and doesn't require many prerequisites. In fact, the author claims

The text is accessible to anyone, from the senior undergraduate to the research scientist. The main prerequisites are the basics of a first course in abstract algebra, the fundamentals of an introductory course in elementary number theory, and some knowledge of basic matrix theory. In any case, the appendices, as described below, contain a review of all of the requisite background material. Essentially, the mature student, with a knowledge of algebra, can work through the book without any serious impediment or need to consult another text.

So my recommendation for the background you should have before you start that book would be the basics of undergraduate group theory, ring theory and linear algebra. The amount that Herstein's Topics in Algebra covers those topics is sufficient. You will be able to appreciate a decent amount of the book with mostly that. Then while you progress through the book, you should also learn about fields and Galois theory.

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Algebraic number theory is a branch of number theory. The algebra is APPLIED to solve number theoretical questions. You do not need to be an expert in commutative algebra to understand it (especially not basic ANT). I feel the ideas and intuitions are more important than the proofs to begin with, there are quite a few interesting studies in this area that can be clouded by details and formalities if studied from purely algebraic sources.

My first text for ANT was Stewart/Tall, "Algebraic number theory and Fermat's Last Theorem". This is a nice book that begins by introducing the basic algebra you need and showing how to use it to study number theory (i.e. solving Diophantine equations by using number fields). They handle the shift from factorisation of elements to factorisation of ideals quite well. The last few chapters handle the applications of this material to Fermat's Last Theorem and goes on to the more recent developments that led to the proof.

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