I know there are algorithms to factor polynomials over finite fields, outlined on this wiki page. I have also found a paper which gives the number of irreducible polynomials of $n$th degree in a finite field of characteristic 2. I am looking for some form of a polynomial of degree $n$ that is irreducible in GF(2) for any $n$. I have briefly considered $x^n+x+1$, but there is a counterexample $x^5+x+1 = (x^2 + x + 1) (x^3 + x^2 + 1)$. I apologize if this is a trivial or well-known result since I have a very poor background in number theory.
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1While counting them is well-known, identifying specific ones in a formulaic (as opposed to merely algorithmic) way for all $n$ is not "well-known to me". :) But more-expert people may know some recent development that are at least semi-formulaic, or anyway more qualitative than any of the obvious more-or-less-efficient algorithms. – paul garrett Apr 14 '21 at 21:18
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Related: General way to find monic irreducible polynomial of degree $n$ and Algorithm to find the irreducible polynomial and Product of all monic irreducibles with degree dividing $n$ in $\mathbb{F}_q$? – 光復香港 時代革命 Free Hong Kong Apr 14 '21 at 23:23
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1For the record, $x^n + x + 1$ is reducible in $\mathbf Z[x]$ (and thus in $\mathbf F_2[x]$) if $n \equiv 2 \bmod 3$ and $n > 2$: $n = 5, 8, 11, 14, \ldots$. These polynomials always have $x^2+x+1$ as a factor. See Example 5 in https://kconrad.math.uconn.edu/blurbs/ringtheory/irredselmerpoly.pdf. – KCd Apr 15 '21 at 01:34
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1For no prime $p$ is there a simple explicit recipe for an irreducible polynomial of degree $n$ in $\mathbf F_p[x]$ for every $n$. – KCd Apr 15 '21 at 01:36