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I’m a freshman math major student and was looking for some recommendations on number theory books. Just to clarify, I was reading Abstract Algebra: An Introduction by Thomas Hungerford.

Although it was my first abstract algebra book, it gave me some background in group and ring theory, modular arithmetic, fields, etc. I was undecided about two books: A Classical Introduction to Modern Number Theory by Ireland and Rosen, and Algebraic Number Theory by Jarvis, but I would like to use only one of them.

At first glance, it seemed that the first one is way more rigorous than the second — specially because it says graduate textbooks, not undergraduate textbooks on it — but it turns out that a lot of people said that this is actually not true, and that the algebraic number theory book by Jarvis is more rigorous.

Because of that, I would like to hear from you opinions on these two books, and I would really appreciate if you could list the prerequisites for each of them. Do I need to know anything besides the basics of abstract algebra to attack these books? Do I need to know some linear algebra, or some analysis or even calculus to handle it?

Moreover, which would be more appropriate — in terms of material covered, clarity and rigor — considering that I have studied abstract algebra using Hungerford's book and that I have a basic (I mean, BASIC) background in elementary number theory? (*) Is there one of them that is more algebraic than the other? Does one of them presents analytic number theory and requires some background in analysis?

I also accept recommendations regarding other books, though I'm really looking for opinions on those two books I mentioned.


(*) I have never used any number theory book before, but Hungerford's book gave me some insights on elementary number theory, and I have attended some college classes in elementary number theory while on my high school senior year.

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  • $\begingroup$ Eventually Ireland and Rosen gets to the some analytic number theory (like chapter 15 or something) but before that I don't think there is much analysis involved $\endgroup$ – D_S Jul 29 '18 at 20:06
  • $\begingroup$ I also remember that Ireland & Rosen eventually requires some Galois Theory. $\endgroup$ – Nagase Jul 29 '18 at 20:17
  • $\begingroup$ Thanks for the comment! I have covered some Galois Theory on Hungerford’s book, but I don’t know if it is enough. However, I don’t know anything of Analysis, so maybe A Classical Introduction to Modern Number Theory is not appropriate for right now. What do you think about Algebraic Number Theory? Is there any Analysis involved ? Is ANT by Jarvis more rigorous than A Classical Introduction to Modern Number Theory by Ireland and Rosen? $\endgroup$ – Guilherme Martins Jul 29 '18 at 20:30
  • $\begingroup$ It seems to me like any book you choose will at some point get into unfamiliar territory with hardly any explanation. But take this with a grain of salt, I have not studied math formally, but more as a dilettante. $\endgroup$ – Robert Soupe Jul 30 '18 at 15:55
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    $\begingroup$ @RobertSoupe I was in fact about to recommend my book (with some modesty) - thanks for doing so. It's inexpensive from Dover with a list of errata (and I get no royalties, so plugging it here is OK). It may be more elementary than what the OP wants (only quadratic number fields) but (I think) a good place to start. And yes I would answer a question about it here. $\endgroup$ – Ethan Bolker Jul 31 '18 at 20:00
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Ireland and Rosen has a lot of great material in it, but it is not a book from which you should learn algebraic number theory: its treatment of that topic only starts near the middle and I felt that I could follow that part only because I had learned algebraic number theory elsewhere already.

The two books you mention are largely on different aspects of number theory, so I think the idea that you can only use one of them is misguided. Read both!

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  • $\begingroup$ Thanks for the comment, KCd. I have decided that I should strengthen my Elementary Number Theory’s background before starting to really study about Algebraic Number Theory. Therefore, I’m really considering to use now A Classical Introduction to Modern Number Theory by Ireland and Rosen and/or An Introduction to Number Theory by Everest before really focusing in ANT - since I have never studied it. With that in mind, what do you think I should use: a Classical Introduction to Modern Number Theory or the other book I mentioned ? Just to clarify it again, I have no experience with Analysis. $\endgroup$ – Guilherme Martins Jul 31 '18 at 22:53
  • $\begingroup$ I also accept other suggestions, but I was looking for a book which cover Elementary Number Theory - but can also cover a little bit of Algebraic Number Theory, as Ireland and Rosen’s book do - with a rigorous approach. Someone has recommended Elementary Number Theory by Burton, but I guess Burton’s approach is more appropriate for those who are dealing with Proofs/Elementary Number Theory for the first time. However, at the same time, I don’t want to take a book which will be so hard that I will not be able to understand it lol. $\endgroup$ – Guilherme Martins Jul 31 '18 at 22:58
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    $\begingroup$ Please refrain from calling books by Ireland-Rosen or Jarvis as non-rigorous or more or less rigorous than each other. While I'm at it, I don't understand people who are asking whether to have salad or potatos with their meat. $\endgroup$ – franz lemmermeyer Aug 1 '18 at 10:59
  • $\begingroup$ @franz your point about not calling books more or less rigorous should be a comment to the original post, not my answer, as I say nothing of the kind. I agree with you that it is not reasonable to suggest because a book is at the graduate level it must be more rigorous than an undergraduate textbook. $\endgroup$ – KCd Aug 1 '18 at 11:02
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I think Ireland and Rosen is a very good choice. It gives an introduction to all parts of number theory with a lot of motivation.

For algebraic number theory in particular, there are many other book recommendations, see here.

Edit: Number theory, even algebraic number theory, will always involve a bit of analysis (geometry of numbers, Minkowski, L-series, class number formula, Dirichlet's unit theorem etc.). One should not be afraid of this.

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  • $\begingroup$ It's worth noting that Jarvis's book wasn't written yet when that question was asked. It's geared to an undergraduate audience, so may be more accessible to the OP than some of the other reccommendations. $\endgroup$ – Mathmo123 Jul 29 '18 at 20:52
  • $\begingroup$ But Ireland and Rosen does both, algebraic number theory in an accessible way, and a motivation for many other areas of number theory. So I would vote for Ireland and Rosen. $\endgroup$ – Dietrich Burde Jul 29 '18 at 20:54
  • $\begingroup$ Of course! Ireland and Rosen gives a broad and motivated introduction to a large number of topics, whilst Jarvis is an algebraic number theory textbook. My comment was aimed at your last sentence: if the OP is after an ANT textbook at their level, they won't find Jarvis's book on that thread, but probably should! $\endgroup$ – Mathmo123 Jul 29 '18 at 21:06
  • $\begingroup$ @Mathmo123 Ah, I see. Yes, this is true. The MO list should be completed by a couple of more accessible books. I just wanted to say, that for the more specialised area of ANT there a also more specialised books. $\endgroup$ – Dietrich Burde Jul 29 '18 at 21:10
  • $\begingroup$ So do you think Jarvis’ book would be more appropriate for my case? If so, do you think that I need to know any Analysis for that (or just Algebra)? $\endgroup$ – Guilherme Martins Jul 29 '18 at 21:49

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