Can we find all the irreducible polynomials of $F_2[x]$?

Question

Can we find all the irreducible polynomials of $F_2[x]$ of a degree $n$?

Is the number of irreducible polynomial of $F_2[x]$ Infinite?

I was to find if there is any degree $n$ such that there is no irreducible polynomial of $F_2[x]$ of that degree.

Can anyone help me by giving some hints? I think my three doubts are related to each other.

Answer 1

The first question is hard. To answer the second and third question:

The Necklace polynomial $M(2,n)$ simultaneously represents:

1. The number of aperiodic necklaces which can be made by arranging $n$ colored beads, from $2$ colors; and
2. The number of monic irreducible polynomials of degree $n$ over a finite field with $2$ elements.

It is clear that this is strictly positive for all $n>0$ (e.g. one white bead and the rest black beads), hence there is at least one monic irreducible polynomial of each positive degree (over $F_2$). In particular, there are infinitely many monic irreducible polynomials (over $F_2$).