Count the primitive polynomials of degree $d$ over $F_{2}$

Question

I am trying to solve an exercise, where i have to determine the number of the primitive polynomials of degree $10$ over the finite field $F_{2}$.

My approach was using the formula $\frac{\varphi (p^{d}-1)}{d}$, where $p$ is the number of the elements of the finite field, $d$ the degree and $\varphi$ is the Euler totient function. So, i get $\frac{\varphi (2^{10}-1)}{10}=\frac{600}{10}=60$.

Do you agree with this solution? I would be glad to read your comments and remarks.

Thanks in advance!

Answer 1

If $q$ is a prime power, then the number of primitive monic degree $d$ polynomials in $\mathbb{F}_q[x]$ is indeed $$ \frac{φ(q^d − 1)}{d}. $$ And primitive polynomials are necessarily irreducible. We have $99$ monic irreducible polynomials of degree $10$ over $\mathbb{F}_2[x]$, and $60$ of them are primitive.