I am new to class field theory, I want to study Hilbert class field for pure cubic fields. Which is the good source? Thank you in advance.

  • $\begingroup$ Can you add some more context to your request? Maybe what material and sources you are already familiar with? What things you have already studied in class-field theory? If you're really "new" to it, then you need to start by understanding Artin reciprocity way before you can study specific class fields. $\endgroup$ – Adam Hughes Sep 1 '14 at 3:55
  • $\begingroup$ Thank you for your reply.I am only familiar with only algebraic number theory now I want to study Hilbert class field theory for pure cubic fields. I want to know good reference book or lecture notes, where in they have explained it with examples(other than quadratic fields). $\endgroup$ – MKJ Sep 1 '14 at 6:10
  • $\begingroup$ "Algebraic number theory" is still a bit broad, that's an entire field. Can you elaborate on what topics you've covered so far? $\endgroup$ – Adam Hughes Sep 1 '14 at 6:14
  • $\begingroup$ I know only ideal factorizations in extensional fields and about class groups. $\endgroup$ – MKJ Sep 1 '14 at 6:34
  • $\begingroup$ Then I would recommend you start by trying to read Serge Lang's algebraic number theory book. It has one of the most basic introductions to class field theory, that avoids most of the heaviest machinery. It should get you through at least Artin reciprocity, and the definition of the Hilbert class field. Perhaps come back again after that. $\endgroup$ – Adam Hughes Sep 1 '14 at 7:02

I don't know any particular source for this.
But if you look at part (i) of Thm.~1.3 of this paper it has a result about the $3$-part of the class group (and hence the Hilbert class field), not for $\mathbb Q(N^{1/3})$, but for the closely related field $\mathbb Q(\sqrt{-3},N^{1/3})$ (where $N$ is a prime satisfying a certain congruence condition).

This paper follows up on the previous one, and proves additional results about the $3$-part of pure cubic fields.

The second paper in particular has some other citations about these sorts of problems in its bibliography. Both papers are fairly recent (about 10 years old), so taken together they may give you at least some sense of the state of the art.

[Side note: the methods of the second paper are pure algebraic number theory. The methods of the first paper are quite a bit more involved; they depend on the theory of deformations of Galois representations that Wiles developed for his proof of FLT. So you may want to look at just the statements of the theorems in the first paper, but not the proofs, while in the second paper (and in some of the other papers it cites) it may be more realistic to try to understand the arguments as well.]


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