Books On Number Theory Pdf
Beichen Poque
Fantastic undergraduate book that covers a lot of ground. While it doesn't require much background, many of the proofs are terse and the authors expect a lot out of you. But it's a very rewarding read and in addition to the number theory you will learn, this book will greatly improve your ability to read mathematics.
The variety of topics covered (...) includes divisibility, diophantine equations, prime numbers (especially Mersenne and Fermat primes), the basic arithmetic functions, congruences, the quadratic reciprocity law, expansion of real numbers into decimal fractions, decomposition of integers into sums of powers, some other problems of the additive theory of numbers and the theory of Gaussian integers.
For Elementary number theory, I would recommend David M. Burton's Elementary number theory and for a bit higher studies my choice is An introduction to theory of numbers by Niven, Zuckerman, Montgomery
From Amazon review: "All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes."
I have just finished a master's degree in mathematics and want to learn everything possible about algebraic number fields and especially applications to the generalized Pell equation (my thesis topic), $x^2-Dy^2=k$, where $D$ is square free and $k \in \mathbbZ$. I have a solid foundation in modern algebra and elementary number theory as well as analysis. Does anyone have any suggestions? I am currently reading Harvey Cohn's 'Advanced number theory' with slow but marked progress. Thanks.
Though Mariano's comment above is no doubt true and the most complete answer you'll get, there are a couple of texts that stand apart in my mind from the slew of textbooks with the generic title "Algebraic Number Theory" that might tempt you. The first leaves off a lot of algebraic number theory, but what it does, it does incredibly clearly (and it's cheap!). It's "Number Theory I: Fermat's Dream", a translation of a Japanese text by Kazuya Kato. The second is Cox's "Primes of the form $x^2+ny^2$, which in terms of getting to some of the most amazing and deepest parts of algebraic number theory with as few prerequisites as possible, has got to be the best choice. For something a little more encyclopedic after you're done with those (if it's possible to be "done" with Cox's book), my personal favorite more comprehensive reference is Neukirch's Algebraic Number Theory.
Marcus's Number Fields is a good intro book, but it's not in LaTeX, so it looks ugly. Also doesn't do any local (p-adic) theory, so you should pair it with Gouva's excellent intro p-adic book and you have a great first course in algebraic number theory.
Many people have recommended Neukirch's book. I think a good complement to it is Janusz's Algebraic Number Fields. They cover roughly the same material. Neukirch's presentation is probably the slickest possible; Janusz's is the most hands on. I love them both now, but I found Janusz understandable at a point when Neukirch was still completely impenetrable.
I would recommend you take a look at William Stein's free online algebraic number theory textbook. It is especially useful if you want to learn how to compute with number fields, but it is still extremely readable even if you skip the details of the computational examples.
There is a fairly recent book (in two volumes) by Henri Cohen entitled "Number Theory" (Graduate Texts in Mathematics, Volumes 239 and 240, Springer). [To avoid any risk of confusion: these are not the two GTM-books by the same author on computational number theory.]
It contains material related to Diophantine equations and the tools used to study them, in particular, but not only, those from Algebraic Number Theory. Yet, this is not really an introduction to Algebraic Number Theory; while the book contains a chapter Basic Algebraic Number Theory, covering the 'standard results', it does not contain all proofs and the author explictly refers to other books (including several of those already mentioned).
Final note: the book is in two volumes, the second one is mainlyon analytic tools, linear forms in logarithms and modular forms applied to Diophantine equations; for the present context (or at least initially), the first volume is the relevant one.
If you want to have a pretty solid foundation of this subject, then you are suggested to read the book Lectures on Algebraic Number Theory by Hecke which is extremely excellent in the discussion of topics even important nowadays, or the report of number theory by Hilbert whose foundation is indeed solid.
In addition, Gauss's book, being a little old and hard, is a good reference on quadratic forms and it itself offers two different kinds of proofs of the quadratic reciprocity law which are all excellent to me.
The last but not the least, I would like to confirm once more the book by Jurgen Neukirch which notes the connection between ideals and lattices, i.e. algebraic numbers and geometry.
I could be wrong, but I think Borevich and Shafarevich cover material related to Pell's equation. If not, then it is still an excellent book on algebraic number theory as is Serre's "A Course in Arithmetic". However Serre does not discuss Pell's equation.
If you want to learn class field theory (which you should at some point, after you have read an introductory book on algebraic number theory), then "Algebraic Number Theory" edited by Cassels and Frhlich is a classic that doesn't get old. It has been recently reprinted by the LMS.
The book Number theory II by Koch (translated by Parshin and Shafarevich) is very good, and contains some hard-to-find material. For example, they give a presentation of the absolute Galois group of a local field.
Also (I can't imagine someone else hasn't suggested this) the conference proceedings Algebraic number theory, edited by Cassels and Frhlich, is a pretty standard text with lots of useful stuff (including Tate's thesis!)
A recent gem is John Stillwell's Algebraic Number Theory for Beginners: A Path from Euclid to Noether (link to publisher's page). It is particularly gentle, even going over the necessary parts of linear algebra. It also does a great job of putting the subject in a historical perspective, as one would expect from Stillwell. Delightful in its own right and a stepping stone to more advanced texts.
Many results in number theory are stated either in a classical language or in an adelic one. I am often impressed of the efficiency and the satisfactory computational properties of the adelic setting, often brief and well-behaved. They seems omnipresent, but barely justified, almost abstruse !
Unfortunately I never found a good reference book for the adeles and ideles definitions and properties : they always are treated in appendix or in a little chapter giving most of the time only the necessary stuff for the self-sufficientness ok the book.
Travelling among tens of lecture notes and books (Weil, Vignras, Goldfeld, Lang, Milne, Tate, Bump, Gelbart, etc. : all books which have not adeles as main theme !) do not seem to be a good solution in order to have a good idea of adelic objects and properties : what are them ? for what do they exist ? are there examples and computation rules ? what are local and global properties ? splitting properties ? measures ? volumes ? general methods ? approximation theorems ? compactness of adelic groups ? are so many questions always only partially answered, often referring to an other book again...
So here is the question : is there any good reference, the more comprehensive possible, starting from the beginning and treating all the major aspects and properties of adeles, but not being just an arid handbook without intuition nor motivation nor examples ?
There is a short chapter constructing general restricted product of groups, giving them their topology and measures, then applying to obtain adeles and ideles groups, plus approximation theorems and others properties, and class group.General but with no computations. - S. Miller, Adeles, Automorphic Forms and Representations (available there)
Karl-Dieter Crisman's Number Theory: In Context and Interactive is a free textbook for an upper-level (US) number theory course, with a clear vision to expose students to the connections to all areas of mathematics. There are many exercises, both proof-based and computational, and nearly every concept can be visualized or experimented with using the open source mathematics software SageMath.
The book tackles all standard topics of modular arithmetic, congruences, and prime numbers, including quadratic reciprocity. In addition, there is significant coverage of various cryptographic issues, geometric connections, arithmetic functions, and basic analytic number theory, ending with a beginner's introduction to the Riemann Hypothesis. Ordinarily this should be enough material for a semester course with no prerequisites other than a proof-transition experience and vaguely remembering some calculus.
The previous major edition was the January 2020, or 2020/1 Edition. Thisaddressed the switch in the Sage cell serverto using SageMath 9.0, which runs on Python 3. Most Sage commands shouldstill work on older versions of Sage; see below for other editions.
The immediately preceding August 2019, or 2019/8 Edition,addressed all known typos/unclear relative pronouns, clarified many proofs,added much more cross-referencing and index entries, and fixed all knownerrata, though it introduced a couple new errata.Other than fixing Sage commands, this edition is identical to the one from 2020.
I've just finished reading Ash and Gross's Fearless Symmetry, a wonderful little pop mathematics book on, among other things, Galois representations. The book made clear a very interesting perspective that I wasn't aware of before: that a large chunk of number theory can be thought of as a quest to understand $G = \textGal(\overline\mathbbQ/\mathbbQ)$. For example, part of the reason to study elliptic curves is to describe two-dimensional representations of $G$, and reciprocity laws are secretly about ways to describe the traces of Frobenius elements in various representations. (That's awesome! Why didn't anybody tell me that before?)
93ddb68554