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Motivation This question came from my efforts to solve this problem presented by Andre Weil in 1951.

Can we prove the following theorem without Axiom of Choice?

Theorem Let $A$ be an integrally closed domain containing a field $k$ as a subring. Suppose $A/fA$ is a finite $k$-module for every non-zero element $f \in A$. Then the following assertions hold.

(1) Every ideal of $A$ is finitely generated.

(2) Every non-zero ideal of $A$ is invertible.

(3) Every non-zero ideal of $A$ has a unique factorization as a product of prime ideals.

EDIT May I ask the reason for the downvotes? Is this the reason for the downvotes?

EDIT What's wrong with trying to prove it without using AC? When you are looking for a computer algorithm for solving a mathematical problem, such a proof may provide a hint. At least, you can be sure that there is a constructive proof.

EDIT Why worry about the axiom of choice?

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    Take an arbitrary statement of non-trivial commutative algebra and ask if it holds in ZF - what's the point? Do you know that in ZF there might be nontrivial rings with no maximal ideals at all? So why should one expect that any "spectral" methods take over?2012-07-12
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    http://math.stackexchange.com/questions/159699/the-group-of-invertible-fractional-ideals-of-a-noetherian-domain-of-dimension-12012-07-12
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    What's spectral methods?2012-07-12
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    See the comments here:http://math.stackexchange.com/questions/169550/existence-of-a-prime-ideal-in-an-integral-domain-of-finite-type-over-a-field-wit/169602#1696022012-07-12
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    I completely agree with Martin. It is very easy to generate a huge amount of these questions all across the board in mathematics. Especially since almost all your questions in this topic are either sparse ("reprove all basic calculus for me.") or you end up writing an answer on your own, it would seem to me that you would *really* benefit from having your own blog where you can post these things in a complete and self-contained format. If you intend to keep posting, every few days, another question of this form and then answer it yourself... then there is something wrong, in my eyes.2012-07-12
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    @AsafKaragila I'm trying to solve this problem:http://math.stackexchange.com/questions/155392/existence-of-valuation-rings-in-an-algebraic-function-field-of-one-variable2012-07-12
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    I think I solved it. But it's long. So I split it to several questions.2012-07-12
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    So it's not arbitrary. Understood?2012-07-12
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    Write a paper and put it on arXiv.2012-07-12
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    I don't think it's worth it.2012-07-12
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    Just because you are not interested in this type of questions does not necessarily mean others are so, too. Please ignore these questions.2012-07-12
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    Why do you think I am uninteresting in this type of questions? In fact I love these sort of questions as they are interesting (research-wise) for me, and to some extent keep the AC related questions in the mathematical interest so it is not becomes a void topic that only a few weirdos care about. What I *do* feel hard to ignore is someone practically turning this website to his blog posting questions and answers that he solved, and making about gazillion edits in the process instead of preparing the answers and questions **in advance** to post them once, and then make final improvements later.2012-07-12
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    What's so wrong in writing answers to my question? My answers are not necessarily correct. If they are wrong, anyone can always correct it. Anyone else can post his answer too. Usually there are several(or many) answers to a question. I've never dreamt my answers were models, nor I'm implying it.2012-07-12
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    @AsafKaragila What's so wrong in editing answers? I asked this many times but you've never answered except you continue to imply it's wrong.2012-07-12
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    @AsafKaragila You don't like my editing style. Everyone has his preference. That's okay, but trying to force your preference to others just because you don't like theirs is not okay.2012-07-12
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    Aren't you forcing me *your* preference by editing so much? Are you not forcing me this just because you don't like to prepare things before posting them? I thought you thought it is not okay to do that.2012-07-12
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    @Asafkaragila No. Please discuss it in meta.2012-07-12
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    Discussion require both sides willing to listen. You have clearly expressed no interest in listening. You wish to complain that people are complaining, but you have no sense whatsoever to what other people dim socially acceptable in this site. How can I discuss this with you, and why would I discuss this with someone else? Especially since if the discussion with someone else will end in agreement that your usage of the site is wrong, you will cry that we are trying to force you just because we don't like it...2012-07-12
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    The use of boldface and triple question marks looks like you are yelling/whining/crying. If you want to ask a question in a civilized way and not look like a pouting child, drop the histrionics. (No, I did not downvote).2012-07-12
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    @ArturoMagidin I'm usually polite, but there are exceptions in my rules.2012-07-12
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    @AsafKaragila Please explain rationally what's wrong with what I'm doing. I think I'm rational enough(meaning not affected by emotion), so there's a chance that you can convince me. However, you've never convinced me so far.2012-07-12
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    @AsafKaragila I'm wating for your reply.2012-07-13
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    @AsafKaragila You continue to say I better write my blog. I disagree. A personal blog can disappear anytime. I believe writing an answer here have more benefits for the members.2012-07-13
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    @AsafKaragila you seem to be so *eager* to discourage me from posting answers to my questions. Is there any particular reason other than you just don't like it?2012-07-13
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    I wonder why some people can't take a mathematical problem just as a mathematical problem. I think whether a questioner knows an answer or not does not matter. As I wrote before, I think whether a problem is homework or not does not matter, either. If an interesting problem is given and I can solve it, I'll post an answer with pleasure no matter whether the questioner has an answer or not, no matter whether it is homework or not.2012-07-13
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    Honestly I hadn't had an answer to this question when I asked it. I came up with an answer only a few days ago. http://math.stackexchange.com/questions/155392/existence-of-valuation-rings-in-an-algebraic-function-field-of-one-variable2012-07-13
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    @Makoto Are you able to accept the answers you post for your questions? This would take them off the unanswered list.2012-07-13
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    @KeenanKidwell I don't know. Is that a problem if I couldn't and it would take them off there?2012-07-13
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    @MakotoKato Well, when you look at the answer that you've posted, is there a check mark under it that you can click? In general there should be I think, although I'm not sure if it's there for unregistered users. If you've answered your question to your satisfaction, then I think you'd want it off the unanswered list since it's specifically for questions which haven't been answered.2012-07-13
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    @Keenan: Just to be clear, in case you were not aware, an edit to a question or answer will still bump a post to the front page of the site regardless of whether there is an accepted answer, so I don't think having Makoto accept his or her own answers will address the main complaint of Asaf and others.2012-07-14
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    @Zev Thanks for the info. I wasn't necessarily endorsing the complaints of Asaf and others. I just thought it would be good for a completely answered question to be clearly indicated as such.2012-07-14
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    @ZevChonoles The main complaint of Asaf Karagila is not that my edits bump my post to the front page of the site. He has never complained it to me.2012-07-15

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