Is there any general way to find 'one' monic irreducible polynomial of degree $n$? Does there exist any algorithm to find it?
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1I'd venture to guess the answer is no. – Rushabh Mehta Nov 08 '19 at 01:21
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3Irreducible over which field? $x^n+2x^{n-1}+\cdots+ 2x+2$ is irreducible over $\mathbb Q$ by Eisenstein's criterion for any $n>0$. – Randy Marsh Nov 08 '19 at 02:00
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I can see now that you have the finite-fields tag. In that case, there are algorithms for computing Conway polynomials, e.g. http://eprints.cs.vt.edu/archive/00000493/ and a database at http://www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol/index.html?LANG=en – Randy Marsh Nov 08 '19 at 02:35
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1Related. – Jyrki Lahtonen Nov 08 '19 at 10:48
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Related 2. – Jyrki Lahtonen Nov 08 '19 at 10:53
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Related 3, 4. – Jyrki Lahtonen Nov 08 '19 at 11:06
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Anyway, a degree $n$ monic polynomial over $\Bbb{F}_p$ is irreducible with probability very close to $1/n$. So with an efficient irreducibility test in place you can find one by random poking soon enough. – Jyrki Lahtonen Nov 08 '19 at 12:55
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(cont'd) But asking for a specific one is asking a lot (apart from the few lists linked to above). The reason for the difficulty is pretty much the same why we don't have a simpe formula that always produces a prime number. The case of irreducible polynomials modulo $p$ is a little bit easier than that of prime numbers (definite version of Rabin-Miller-Fermat style tests, the analogue of Riemann hypothesis is a theorem, etc), but still. – Jyrki Lahtonen Nov 08 '19 at 12:58