Hopefully this is not too easy an exercise.

Let $F$ be a field. Let $I \subset \mathbb{N}$ be the set of all positive integers $d$ such that there exists an irreducible polynomial of degree $d$ over $F$. What kind of $I$ can occur?

Of course $1 \in I$, and of course we can have $I = \mathbb{N}$ or $I = \{ 1, 2 \}$ or $I = \{ 1 \}$. The Artin-Schreier theorem implies (I think) that if $I$ is finite, then only the latter two cases occur. So what kind of infinite $I$ can occur?