Now I'm reading Mollin's "Algebraic Number Theory" and I'm confused with some of author's argument.

  1. When this book proving every ideal of Dedekind domain is invertible, it just prove when ideal is principal, and says general case can be done by induction. But is there any way to prove this induction step simply?

  2. This book says ring of integer is Noetherian because of definition number field and $\mathbb{Z}[\alpha]$ is finitely generated module when $\alpha$ is integral. Is it enough? I don't think so..

In both case I have no idea. In other books I read, like Neukirch's and Milne's algebraic number theory book, they use longer argument for 1 (and they use `induction' in quite different way..), and use trace to prove ring of integer is finitely generated $\mathbb{Z}$-module. Is there exist simpler way? I'm too stupid or author's argument is not enough, which one is right?


Concerning 2., Neukirch's proof is explained at this MSE question: Rings of integers are noetherian (question about a specific proof). I think, this proof is fine. You are asking "Is there a simpler way"? Basically, no, but this depends on the prerequisites we have. I do not agree that "trace" is not "simple" enough.

Concerning 1., I prefer the proof given in Milne's lecture notes. I cannot see why this proof is not fine. For comparing with Mollin's proof, could you give a link?

  • $\begingroup$ I'm fine with Neukirch's and Milne's proof. But in Mollin proof is too short and seems insufficient so I'm wondering is there a proof that justify Mollin's argument. Other two author's proof is far longer and so I don't think Mollin intend those proves.. $\endgroup$ Jul 27 '17 at 9:51
  • $\begingroup$ "Justify Mollin's argument" - which argument? I think we really need to read all of Mollin's text first. $\endgroup$ Jul 27 '17 at 9:53
  • $\begingroup$ Maybe it's my mistake that didn't explain what is exactly Mollin's argument. For 1, he first prove every principal ideal is invertible(which is trivial), and says "Now the result may be extrapolated by induction, and the result is established." For 2, "By Definition 1.29 and Claim 1.3 in the proof of Theorem 1.22, we know that for any number field $F$, $O_F$ is finitely generated as a $\mathbb{Z}$-module." Def 1.29 is definition of number field, and claim 1.3 is "If $\mathbb{Z}[\alpha]$ is finitely generated module when $\alpha$ is integral" $\endgroup$ Jul 27 '17 at 9:58
  • $\begingroup$ Both proves are just one-sentence and other proves from other authors seems too long for this one-sentence argument' so I wonder is there a shorter proof that fits this one-sentence argument' . $\endgroup$ Jul 27 '17 at 10:00

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