Recently, Arul asked http://mathoverflow.net/questions/139313/ , which struck me as a reasonable question, if a bit under-specified, for a professional mathematician non-number-theorist to ask of number theorists. I had a ton of remarks (though no perfect answer) which I started to leave in comments but, after realizing they wouldn't fit, decided to leave as a CW answer. At the time, I saw that the question had one close vote. While I was writing, Arul deleted the question. I was annoyed at losing what I had written, so I recopied the question and posted it at http://mathoverflow.net/questions/139328 together with my answer. Then Arul un-deleted his question. There is no point in having two copies, so I have reposted my answer on Arul's version (CW) and will shortly delete my question.

However, I had a long conversation with Felipe Voloch in the comments to my version, which I'd like to be able to refer to if there is further discussion about Arul's question. So I am archiving it here with Felipe's permission. I don't have anything more I want to say, but certainly Felipe can feel free to get in the last word here if he wants to.

This is not MO level. Factor and compute the Galois group. – Felipe Voloch 56 mins ago

@FelipeVoloch It seems to me there is a lot to say here. I was going to leave this in an answer, which is taking some time to write. Basic points: – David Speyer 53 mins ago

Although factorization of polynomials over ℚ is rapid, computation of Galois groups is not. Testing whether a Galois is abelian should be much faster than running a general Galois computation algorithm on something abelian. – David Speyer 52 mins ago

Once you know that the Galois gorup is abelian, finding the cyclotomic field is tricky. Matt Emerton started to write out the details here math.stackexchange.com/a/36664/448 and didn't give them all. – David Speyer 50 mins ago

Before calling GAP (for example) to compute the Galois group, there are some obvious Frobenius element plausibility tests which will allow us to reject almost all f. – David Speyer 50 mins ago

"rapid" was not specified in the question. If you want something faster, you can compute the discriminant and narrow down the choice of cyclotomic field, or a number of other things, depending of exactly you want. But that won't make it research level. – Felipe Voloch 48 mins ago

Calegari-Scott-Morisson arxiv.org/abs/1004.0665 have some interesting results when the root is small. – David Speyer 48 mins ago

It is research level in the sense of a computational problem which can reasonably come in research and which a graduate student shouldn't be expected to know how to do. (Agreed that it is not research level in the sense of something you could publish a paper on.) – David Speyer 47 mins ago

When someone asks "how to decide" something, I think it should be taken as the default understanding that they mean in practice. – David Speyer 45 mins ago

It seemed to me more like a random question than that the person actually had a number that he/she needed to decide whether it was cyclotomic. I don't think it's up to you to revive and/or reinterpret the question. If a downvote caused them to delete the question, they don't need the answer that badly. – Felipe Voloch 41 mins ago

I mostly revived it because I was annoyed that I had 80% of the below written out when the question disappeared. I wish the question had had more context (and wish the close-voter had requested some), but otherwise I think it is a good example of the sort of question where experts in computational number theory (not that I am claiming to be one) could quickly help someone who is doing reasonable research in a different field. – David Speyer 39 mins ago

I originally started writing the below in the form of a comment asking for the question to get more specific but, as often happens, it didn't fit in the comment box. Note also the bolded comments as places where an expert (like Felipe!) could probably say something better. – David Speyer 37 mins ago

@FelipeVoloch "It seemed to me more like a random question than that the person actually had a number that he/she needed to decide whether it was cyclotomic" I do have some numbers I want to decide to be cyclotomic. If they are not, then I would know either some natural limits of my technique or the known bound is loose (with relevancy to a problem I am working on). – Arul 9 mins ago

On a purely practical note (not sure if better or worse); you could have asked (or still could) for your question to be migrated to (new) meta; then also timestamps and so on mentioned by Asaf Karagila, would be preserved.