The question is from Marcus' book, "Number Fields" (exercise 12, Chapter 4)
Let $\omega= e^{\frac{2\pi i}{m}}$ and $p$ be a rational prime not dividing $m$. Then how does $p$ split in $\mathbb{Q}[\omega + \omega^{-1}]$?
I know that $\mathbb{Q}[\omega + \omega^{-1}]$ is subfield of $\mathbb{Q}[\omega]$ and that $\text{Gal}(\mathbb{Q}[\omega]/\mathbb{Q})$ is isomorphic to $\mathbb{Z}_m^{\times}$.
The following result is known:
Lemma: Let $H$ be the subgroup of $\mathbb{Z}_m^{\times}$ fixing a subfield $K$ of $\mathbb{Q}[\omega]$ pointwise and $p$ be a rational prime not dividing $m$. Then $f$ is the inertial degree of any prime $\mathfrak{P}$ of $K$ lying over $p\mathbb{Z}$, where $f$ is the least positive integer such that $p^f \equiv x \pmod m$ with $x\in H$.
Since $\mathbb{Q}[\omega]$ has degree 2 over $ \mathbb{Q}[\omega + \omega^{-1}]$, I get $H = \mathbb{Z}_3^{\times}$ and have to find least possible values of $f$ such that $p^f \equiv 1,2 \pmod m$.
What can I conclude from all this, about the splitting of primes?