In a recent theorem we have naturally come across this condition, that seems to be important, but rarely satisfied: 
$\sqrt{\frac 1 4 + a^m} \in \mathbb Q(\zeta_m, a)$
where $a\in\mathbb C^*$ and $\zeta_m$ is a primitive $m$-th root of unity. 

We are especially interested in the case where $a$ is a root of unity. We have checked that the containment condition is fulfilled, whenever $a^m=1$ and $5\mid m$ as this reduces to $\sqrt 5\in\mathbb Q(\zeta_m,a)$ or $a^m=-1$ and $3\mid m$ as this reduces to $\sqrt{-3}\in\mathbb Q(\zeta_m, a)$. Numerically these seem to be the only examples at $a$ root of unity. 

Is there a natural way to see this using algebraic number theory?