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I want to read Lawrence Washington's An Introduction to Cyclotomic Fields. However, my knowledge of algebraic number theory doesn't extend farther than what's found in A Classical Introduction to Modern Number Theory - Ireland and Rosen. Does anyone know whether this will be sufficient or whether I'll have to learn more about algebraic number theory before I can get to it?

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    $\begingroup$ My suggestion: start reading Washington. You will come across notions you will have to study: characters, local fields, analytic methods etc. In my opinion it is not a good idea to wait until you have covered everything you need. $\endgroup$
    – user23365
    Oct 12, 2017 at 14:48
  • $\begingroup$ @franzlemmermeyer Thank you for your reply. I think I'm gonna do that. I've been postponing to read this book and others on my wish list (your one on reciprocity laws as well as Nancy Childress' one on Class Field Theory) despite having a good background in algebra. $\endgroup$
    – J. Doe
    Oct 13, 2017 at 3:06

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I think you'd be happier to have a full introduction to algebraic number theory, like S. Lang's, or equivalent. Then you'll be able to see many of the features of cyclotomic fields as special cases of what would happen more generally, rather than having those special cases appear as novelties.

That is, I think it is useful to see the basic features of cyclotomic fields appear as especially accessible examples of general algebraic number theory, rather than as first-encounter extensions of the number theory of $\mathbb Z$ and quadratic extensions, for example.

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The preface to the first edition of Washington's book says:

The reader is assumed to have had at least one semester of algebraic number theory (though one of my students took such a course concurrently). In particular, the following terms should be familiar: Dedekind domain, class number, discriminant, units, ramification, local field. Occasionally one needs the fact that ramification can be computed locally. However, one who has a good background in algebra should be able to survive by talking to the local algebraic number theorist. I have not assumed class field theory; the basic facts are summarized in an appendix. For most of the book, one only needs the fact that the Galois group of the maximal unramified abelian extension is isomorphic to the ideal class group, and variants of this statement.

It seems you'll have to learn much more about algebraic number theory than covered by Ireland and Rosen.

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