I was playing around with the Frobenius map, and made a small observation.
Suppose $F$ is a field, and $F[X,Y]$ is the corresponding polynomial ring in two indeterminates. If $\text{char}(F)=p$ divides some integer $n$, then $n=pq$ for some $q$. Consider the polynomial $X^n+Y^n+1$. Then $ X^n+Y^n+1=X^{pq}+Y^{pq}+1^{pq}=(X^q+Y^q+1^q)^p $ so $X^n+Y^n+1$ is reducible. I hope this observation is correct.
Does the converse also hold? That is, if $X^n+Y^n+1$ is reducible, can you conclude that $\text{char}(F)$ divides $n$? I tried finding some factorization, and I think it should look something like $(X^r+\text{stuff}+1)$ and $(X^s+\text{stuff}+1)$ where $r+s=n$, but couldn't actually find an explicit factorization.