Projective transitivity of a trinomial linearized polynomial ($L(X)=X^{q^d}+X^q+TX$)

For $d\ge 2$, nonzero roots define projective-point coordinates $s=X^{q-1}$, satisfying $s^N+s+T=0$, where $N=(q^d-1)/(q-1)$. This polynomial is irreducible over ${\mathbb{F}}_q(T)$. After fixing one root $s$, the remaining coordinates satisfy $R(Y,s)=(Y^N-s^N)/(Y-s)+1$. Substitute $z=Y/s$ and $h=1/s$: the equation becomes $h^{N-1}+(z^N-1)/(z-1)=0$. At a nontrivial $N$th root of unity the second term has a simple zero, so the polynomial is Eisenstein over the algebraic closure's rational function field. This proves irreducibility of $R$ and hence two-transitivity on projective points.