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I have Lang's 2 volume set on "Cyclotomic fields", and Washington's "Introduction to Cyclotomic Fields", but I feel I need something more elementary. Maybe I need to read some more on algebraic number theory, I do not know.

So I would appreciate suggestions of books, or chapters in a book, lecture notes, etc. that would give me an introduction. I am specifically interested in connection of cyclotomic fields and Bernoulli numbers.

Thank you.

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up vote 8 down vote accepted

I would just start by looking at Marcus' Number Fields for the basic algebraic number theory. It also contains tons of exercises. If you read the first 4 chapters, you should have the necessary background for most of Washington's book. I'm not familiar with Lang.

I started studying algebraic number theory last summer by going through Marcus book.

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pki's suggestion is good. A couple of other books worth a look are Pollard and Diamond, The Theory of Algebraic Numbers (in the MAA Carus Mathematical Monographs series), and Stewart and Tall, Algebraic Number Theory. Ireland and Rosen, A Classical Introduction to Modern Number Theory, doesn't get as far into algebraic number theory as the others, but it is well-written and has a chapter on cyclotomic fields and a chapter on Bernoulli numbers.

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The book by I&R does not prove the unit theorem or give geometry of number methods for the finiteness of the class group (they use a different approach to bound the class number) but I think other than that it's a more comprehensive volume that Pollard and Diamond. I'd suggest looking at Samuel's Algebraic Theory of Numbers, which does have a section on the cyclotomic field generated by $p$th roots of unity for prime $p$. –  KCd Nov 7 '11 at 5:29
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