# Book(s) Request to Prepare for Algebraic Number Theory

I would appreciate suggestions for books to enhance my learning in algebra so as to be able to read Samuel's "Algebraic Theory of Numbers" and eventually at least begin Neukirch's "Algebraic Number Theory."

By way of background, I have gone through B. Gross's Harvard lectures on algebra several times. They were correlated to Artin, and included factoring and quadratic number fields, but did not cover modules or fields. Nor Galois Theory.

So I would like to get a good exposure to those areas that are particularly germane to ANT. (E.g. Artin does not have anything on perfect fields and only mentions algebraic closure in a short paragraph before the Fundamental Theory of Algebra.

Thanks very much.

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If you want a good, free, online source for abstract algebra, including basic module and field theory, as well as some galois theory, I recommend Robert Ash's course: math.uiuc.edu/~r-ash/Algebra.html – M Turgeon Jun 2 '12 at 20:08
@MTurgeonThanks looks good. – TheBirdistheWord Jun 2 '12 at 20:30

I'll convert this to an answer, since it's getting somewhat long and I believe it answers the question.

Artin's book is more than enough preparation for Samuel's.

Maybe I can allay your fears about what Artin omits. In basic algebraic number theory your fields are either of characteristic zero (finite extensions of $\mathbf Q$ and their completions) or are finite fields (reductions of rings of integers modulo primes), and these are always perfect. Infinite extensions do not play a significant role until you start learning class field theory.

Note too that Samuel defines and proves basic properties about principal rings, modules, algebraic extensions, Galois extensions, and more; so in theory you wouldn't have to know much about those in order to begin reading. It's a very approachable book.

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@DylanMorelandThanks, glad you did. – TheBirdistheWord Jun 2 '12 at 20:29
@Andrew Didn't take so long. I'd like to edit it such that it's clear that I'm not suggesting you skip learning about Galois theory and so on from other books: any time you spend with Artin's book now will probably pay off later. Other good sources are Milne's notes and the book by Morandi. The most important thing is that you're closer than you think to a lot interesting mathematics. Good luck! – Dylan Moreland Jun 2 '12 at 20:35
Also, you might want to avoid accepting an answer for a while. I'm sure there will be other answers from people far more qualified to talk about this than I, and you might find some good references. – Dylan Moreland Jun 2 '12 at 20:41
@Andrew You had a comment on here earlier about sources which I had meant to respond to earlier. How's this going? – Dylan Moreland Jun 16 '12 at 9:07
@DylanMorelandHow kind of you to ask. I swear I was about to send you a message thanking you for your encouragement. I had really gotten stumped on the section about norms and traces. But just stuck with it for many days with your advice in mind. I'm sure you know the delight when you finally figure it out. Hope you're having a great summer. Best regards. – TheBirdistheWord Jun 17 '12 at 11:19

Well, if anyone's interested, I started "Ireland and Rosen." As many people have pointed out (even as recently as a few minutes ago) it's a great book which I think will be beneficial in it's own right, and as preparation for the future.

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Pierre Colmez has written an amazing text, entitled Éléments d'analyse et d'algèbre (et de théorie des nombres). Here's the table of contents. It's so technically precise and well digested that I think probably this book could serve as a nice reference for most of undergraduate analysis and algebra as well. This text together with Samuel's book should be good preparation. There's also Dino Lorenzini's An invitation to arithmetic geometry, which I personally like quite a lot.

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I didn't know about this book by Colmez. Thanks! – Dylan Moreland Jun 28 '12 at 23:05
@DylanMoreland You're welcome! – vgty6h7uij Jun 28 '12 at 23:11
Hm. Can't find it in the library or online, which is too bad. In theory, he will have a lot of interesting things to say. – Dylan Moreland Jun 28 '12 at 23:16
@DylanMorelandHi Dylan. This might work:amazon.com/elements-danalyse-dalgebre-theorie-nombres/dp/…. Best, Andrew – TheBirdistheWord Jun 28 '12 at 23:34