# Error in Mollin's "Algebraic Number Theory"?

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?

• 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" Jul 27 '17 at 9:58
• 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' . Jul 27 '17 at 10:00