The number of irreducible polynomials of degree $4$ over $\FF_2$

Let \(N_q(n)\) denote the number of monic irreducible polynomials of degree \(n\) over \(\mathbb F_q\). From \[ x^{q^n}-x=\prod_{d\mid n}\prod_{\substack{f\text{ monic irreducible}\\ \deg f=d}}f(x) \] we get \[ q^n=\sum_{d\mid n}d\,N_q(d). \] Möbius inversion gives \[ N_q(n)=\frac1n\sum_{d\mid n}\mu(d)q^{n/d}. \] Hence \[ N_2(4)=\frac14\bigl(2^4-2^2\bigr)=3, \] because \(\mu(4)=0\).

Thus there are exactly \[ \boxed{3} \] monic irreducible quartics over \(\mathbb F_2\).