For each odd prime $p$, choose an arbitrary $x\_p \in \mathbb Q(\zeta\_p) \smallsetminus \mathbb Q$, where $\zeta\_p = e^{\frac{2\pi i}{p}}$. Is it always true that the set $\{x\_p : p \in \mathbb P\}$ is linearly independent over $\mathbb Q$?



Cyclotomic fields and $Q$-linearly independence
I think everything is clear now by George Lowther's [comment](https://mathoverflow.net/questions/61123/cyclotomic-fields-and-q-linearly-independence#comment153670_61123), reproduced here with two minor typos corrected:



> 
> $\mathbb Q(\zeta\_m) \cap \mathbb Q(\zeta\_n) = \mathbb Q$ also follows from $[\mathbb Q[\zeta\_m] : \mathbb Q] = \varphi(m)$ and multiplicativity of Euler's totient function $\varphi$.
> 
> 
> 


