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It only takes a minute to sign up. I have the privilege of teaching an algebraic number theory course next fall, a rare treat for an algebraic topologist, and have been pondering the choice of text. The students will know some commutative algebra, some homological algebra, and some K-theory. The texts I am now considering are. I am leaning toward 1 : it seems very well written and has sections on cubic, biquadratic and sextic fields as well as quadratic and cyclotomic fields.
I do know that when you get into a course you discover drawbacks that aren't apparent when you're reading the text before the course begins. Frohlich and Taylor is designed for a year long course and I only have one semester, so will need to do some selecting. I regret not using any homological techniques, also.
There is a short pages , well-written and dense book on the subject : Algebraic Theory of Numbers, by Pierre Samuel translated from French by Allan J.
Very good for a semester course. I'm a big fan of Milne's, however, I'm finding Froehlich-Taylor currently the most balanced mid-level text book available. That being said, if I were you I'd have a good look at H. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambridge University Press, , which manages to be so wonderfully efficient, while at the same time, quite like F-T, it also brings plenty of relevant examples a whole chapter 3.
Janusz certainly merits a place as a classic of class field theory, but is overall perhaps not as student-friendly as the other two. On that note, Algebraic Numbers by Paulo Ribenboim Wiley-Interscience is extremely gentle with the reader and driven by concrete problems. In my opinion, the book "Algebraic number theory" by Serge Lang deserves to be mentioned in this list, as well.
My favorite textbook on the subject is "Number fields" by Marcus. Things are done in a very explicit way, there are hundreds of exercises, and I've usually found that the proofs of the classical theorems finiteness of the class number, structure of the units, I think the book Algebraic number theory by Helmut Koch should be mentioned too, together with his book Number Theory: Algebraic Numbers and Functions.
The online lecture notes of Milne are also excellent, in my opinion, and contain the theory together with many examples. It's not entirely focused on algebraic number theory which is the subject of the last part of the book, but almost no prerequisite is needed. The author introduces all necessary elements about Galois theory and whatever might be needed to start studying algebraic number theory.
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Viewed 4k times. I would appreciate suggestions, either about these or other texts. Robert Bruner. You should also look at Neukirch's book. I fear that I don't have the dexterity to handle it well. On the other hand, i still remember the delight I felt when I first sat down with Neukirch.
Strangely enough, I actually prefer the line-by-line writing in Janusz, though it is less "exciting". But probably, as you say, it is more prudent to choose Frohlich-Taylor. Active Oldest Votes. Best of luck with your course! Salvatore Siciliano. Laurent Berger. It's great otherwise.
Algebraic Number Theory
In the spring semester I organized meetings to answer questions and lecture on the background for Prof. Zhang's course on class field theory. We met every Friday in Math Local Fields : Here are some notes on local fields. I lectured on these during our first two meetings. These notes also contain useful references.
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Algebraic Number Theory by A. Fröhlich