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How do mathematicians know that what they are researching has not been already known for $200$ years? Obviously, if they are researching something that is cutting edge it is not a problem, but if one is investigating a problem in a very old field like Euclidean Geometry then this could be a problem.

I am interested in the problem, how many prime polynomials of degree $n$ are there in $\mathbb Z/p\mathbb Z[x]$? When I google this problem, I get no relevant results. However, for all I know, Gauss solved this problem. But how do I find out?

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    $\begingroup$ They don't know. They talk to other mathematicians, and hope if it was found previously, or if an approach was taken before, somebody might know about it. But they never really know for sure, and sometimes it doesn't matter, if the result has descended into obscurity. $\endgroup$ – Thomas Andrews Jul 29 '14 at 16:36
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    $\begingroup$ The best approach would be to go to a library and spend some hours checking some books. Discussing with some experts in the area is the other option. $\endgroup$ – N. S. Jul 29 '14 at 16:37
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    $\begingroup$ Presumably you mean primes in $\mathbb Z/p\mathbb Z[x]$. This is a well-known problem with a well-known solution. $\endgroup$ – Thomas Andrews Jul 29 '14 at 16:37
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    $\begingroup$ How do you know what is known? Not in mathematics, in general. $\endgroup$ – Asaf Karagila Jul 29 '14 at 16:38
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    $\begingroup$ If I understand your question correctly, the problem was solved by Gauss. Number of monic irreducible polynomials of degree p over finite fields $\endgroup$ – MJD Jul 29 '14 at 16:41
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Is this question still open? Often times, mathematicians don't know what is known. I published a paper based on an "unsolved problem" that someone had posed. It turned out that it had already been solved 40 years ago, but it had been forgotten. But after my paper was published, someone remembered the old paper (it wasn't in Math. Reviews as it had been in a small European journal). This happens a lot. Usually you talk to other people in the area first, before doing any serious work on a problem .

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If you don't have a network of colleagues to ask, you can always just ask on this SE site or on Math Overflow. This is a fairly common practice; if you search for the phrase "Is it known" you'll find several hundred hits (although some of these are false positives).

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