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I found that $x^2+x+1$ is the only polynomial to the power 2 that is irreducible in $z_2$.

Moreover I found that $x^3+x+1$ and $x^3+x^2+1$ are the only polynomials to the power 3 that are irreducible in $z_2$.

Finally I found that that $x^4+x+1$, $x^4+x^2+1$, $x^4+x^2+1$ and $x^4 +x^3+x^2 +x+1$ are the only polynomials to the power 4 that are irreducible in $z_2$.

Hence suppose I want to find an irreducible polynomial, whose largest power is n, in $z_2$. Is there a formula which would let me know how many of these there are?

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marked as duplicate by Jack D'Aurizio, Elaqqad, k170, A.P., Gabriel Romon May 2 '15 at 19:42

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  • $\begingroup$ It might help arxiv.org/abs/1001.0409 $\endgroup$ – hyperkahler May 2 '15 at 15:21
  • $\begingroup$ Maybe it is the case to replace $z_2$ with $\mathbb{F}_2$, just to make clear we are dealing with irreducible polynomials over the finite field with two elements. $\endgroup$ – Jack D'Aurizio May 2 '15 at 15:22

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