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?

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 interested in an equivalent condition for any $a$, but we learned here that the condition for some $a$ seems to be just a technical obstacle of our method of proof and not intrinsic to the problem.

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 and they all lead to different results in our theorem. So we believe that the condition is natural.

Is there a way to prove that these conditions are equivalent to the containment condition for $a$ root of unity - maybe using algebraic number theory?

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.

From a different proof of our theorem, we expect that for $\left|a\right|<\frac 1 4$ this containment is impossible. Is there a natural reason for this?

As a follow-up question: 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 $\sqrt 5\in\mathbb Q(\zeta_m,a)$. 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?

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?