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I am about to start learning Algebraic Number Theory and thus was looking for some expository articles on this subject. So far I have found two such articles:

  1. Dickson, L. E.. (1917). Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers. Annals of Mathematics, 18(4), 161–187. http://doi.org/10.2307/2007234
  2. Mazur, Barry. (2008). Algebraic Numbers. has appeared as a chapter in the book The Princeton Companion for Mathematics, by Timothy Gowers, June Barrow-Green, Imre Leader (Editors), American Mathematical Society (2008). http://www.math.harvard.edu/~mazur/preprints/algebraic.numbers.April.30.pdf

I wish to read more such articles before I start solving Marcus's book. So, I will be happy to know about more such articles.

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  • $\begingroup$ I assume you took to heart Mazur's recommendation of Gauss and Davenport. $\endgroup$ – roy smith May 17 '16 at 15:10
  • $\begingroup$ @roysmith yes it is so. $\endgroup$ – rationalbeing May 17 '16 at 16:52
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http://www.jstor.org/stable/2317083

This article is great! It uses reciprocity laws as a guiding theme through some basic ideas of class field theory, in an attempt to 'go back and figure out the number theory that lay behind all those cohomology groups.'

What is a Reciprocity Law?

B. F. Wyman

The American Mathematical Monthly

Vol. 79, No. 6 (Jun. - Jul., 1972), pp. 571-586

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    $\begingroup$ Yes, it's nice article. I have a curiosity : "Why do most 1 semester courses on Algebraic number theory end before getting into class field theory?". Can you satisfy this curiosity? $\endgroup$ – rationalbeing May 20 '16 at 15:21
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    $\begingroup$ @rationalbeing, a one-semester course already has so much material to cover that it is usually not feasible to reach class field theory except maybe to say briefly something about the main theorems; certainly there's no time to "get into" class field theory in depth. If you take an algebraic number theory course, or later try to teach such a course, you'll understand why. Your question is sort of like asking why a first calculus course doesn't get to Stokes' theorem. $\endgroup$ – KConrad May 20 '16 at 15:25
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    $\begingroup$ my own undereducated guess might be that it's more like 'algebraic number theory' as the body of literature/techniques/problems, with class field theory at/as its heart (see also the first book of the number theory trilogy by Kato et al., books.google.com/…) $\endgroup$ – Samantha Y May 20 '16 at 18:59
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    $\begingroup$ @rationalbeing, just google all the books on algebraic number theory you can find and look at the table of contents before they reach class field theory (many don't even get to that point). You'll see what is covered. I would just call it algebraic number theory. It doesn't have another standard name. Highlights include unique factorization of ideals, the ideal class group, the unit theorem, ramification (discriminant/different), features of Galois extensions (decomposition/inertia groups, Frobenius elements), and some examples (e.g., quadratic and cyclotomic fields) and applications. $\endgroup$ – KConrad May 20 '16 at 20:04
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    $\begingroup$ Not only is the first title long, but it's awkward in English. (I can't judge how the analogue of it in your native language would sound, or maybe you are going to write it in English.) The second one definitely sounds better. $\endgroup$ – KConrad May 21 '16 at 3:02
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I would recommend "Fermat's Last Theorem A Genetic Introduction to Algebraic Number Theory" by Harold Edwards: http://www.springer.com/us/book/9780387902302 See also the first answer to this question https://math.stackexchange.com/questions/387705/preparations-for-reading-algebraic-number-theory-by-serge-lang

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  • $\begingroup$ I asked my guide about this book by Edwards a couple of months ago and he plainly rejected to comment and asked me to solve Marcus's book. Have you read this book by Prof. Edward? Can you give your personal opinion/review about this book? $\endgroup$ – rationalbeing May 20 '16 at 12:13
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    $\begingroup$ How do you "solve" a book? $\endgroup$ – Gerry Myerson May 20 '16 at 12:35
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    $\begingroup$ Here books.google.ru/… this book is called "marvelous". Another, not so favorable review, is here: projecteuclid.org/euclid.bams/1183548007 I read some sections of the book many years ago as a amateur and liked it. $\endgroup$ – Zurab Silagadze May 20 '16 at 12:42
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    $\begingroup$ @GerryMyerson I meant to solve the exercises in the book. $\endgroup$ – rationalbeing May 20 '16 at 15:09
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    $\begingroup$ @rationalbeing Marcus' book is very rich in instructive exercises, except for one big problem: his avoidance of the basic concept of localization (even just for integral domains) at a prime ideal or multiplicative set causes too many of his proofs and definitions to be very "global" in nature, preventing the possibility of instantly reducing certain tasks to the case of dvr's where various matters simplify tremendously. For this reason, I recommend to first study Pierre Samuel's book "Algebraic Theory of Numbers", solve many exercises there, and use Marcus' book for more practice. $\endgroup$ – nfdc23 May 21 '16 at 3:10

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