A minor note: it suffices to limit oneself to closure under roots of polynomials of form $x^n + b$ or $x^n + x + b$, since any other polynomial of form $x^n + ax + b$ can be transformed into the latter by the change of variables $x = a^{1/(n-1)}y$ (and $a^{1/(n-1)}$ is 'available' by virtue of the former).