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While studying "Number Fields" by Marcus, I got stuck with this question, which is question 29 in exercises of chapter 5. And it says the following.

a) Prove that every finitely generated ideal in $\Bbb A$ is principal. b) Find an ideal in $\Bbb A$ which is not finitely generated.

Here, $\Bbb A$ denotes the ring of all algebraic integers in $\Bbb C$. I have no idea how to deal with part a (a hint would be great!). For part b, for a fixed $a$ the ideal generated by the elements $\sqrt[n]{a}$ for all $n\in \Bbb N$ seems to me a correct example. Because if it's finitely generated, then for a prime $p$ greater than the maximum $n$ of $\sqrt[n]{a}$ inside the generating set, $\sqrt[p]{a}$ wouldn't be generated by the finite generating set. However since it seems a bit trivial, I'm afraid I may have missed a point. Any hint for both parts is welcomed.

offret
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  • You might want to read this question. (a) is the same as saying that $\mathbb{A}$ is a Bezout domain, and the accepted answer to the linked question gives a reference for a proof of that fact. – rogerl Jun 02 '19 at 20:41

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This question and various refinements of it have been asked several times before on this site (e.g. here)

Hint for (a): Let $I$ be a finitely generated ideal in $\mathbb A$, and write $I=(\alpha_1,\dots, \alpha_n)$. Let $K$ be the number field generated over $\mathbb Q$ by the $\alpha_i$ and denote by $\mathcal O_K$ the ring of integers of $K$. Let $J$ be the integral ideal of $K$ generated by the $\alpha_i$. Since the class number of $K$ is finite there is some positive integer $m$ such that $J^m$ is principal. Let $\beta\in\mathcal O_K$ be such that $J^m = (\beta)$. Can you finish it up from here?

Hint for (b): I think that the approach you outline in your question should work out. In particular you could try to argue that if your ideal was finitely generated then all of the $\sqrt[n]{a}$ would have to lie in some finite degree extension of $\mathbb Q$, which they obviously do not.

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    Very much appreciated! Part (a) seems trivial now, and the idea for (b) seems reasonable, I'll try to write down the full proof. Thanks! – offret Jun 02 '19 at 20:59