# Is there a formula which would let me know how many irreducible polynomials there are to the power n, in $z_n$? [duplicate]

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?

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

• 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. – Jack D'Aurizio May 2 '15 at 15:22