Is $\mathbb{Q}(z,\dots,z^{n-1})$ the splitting field of some polynomial $\mathbb{Q}[x]$, where $z$ is a primitive root of unity?

Question

Is $\mathbb{Q}(z,...z^{n-1})$ the splitting field of some polynomial in $\mathbb{Q}[x]$, where $z$ is a primitive root of unity?

I know that if $n\ge 1$ and $k$ is a field, and $f(x)=x^n-1\in k[x]$, then there is an extension $K/k$ over which $f(x)$ splits. But I am not sure if this is even relevant to the question at hand.

Thanks.