Splitting of primes in real cyclotomic field 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?
 A: Based on hints by Jyrki Lahtonen :
First of all, $H=\{\overline{1}, \overline{-1}\}$ (which is not same as  $\mathbb{Z}_3^{\times}$). So, using the Lemma stated in question I can calculate $f$ to be the least positive integer such that $p^f \equiv \pm 1 \pmod m$.
Since $p\not|m$, by a standard theorem for $\mathbb{Q}[\omega]$, we know that its ramification index is 1. Moreover, $\mathbb{Q}[\omega + \omega^{-1}]$ is a subfield of $\mathbb{Q}[\omega]$, hence we conclude that its ramification index is also 1. 
We know that $\mathbb{Q}[\omega]$ has degree 2 over $\mathbb{Q}[\omega + \omega^{-1}]$ and degree $\phi(m)$ over $\mathbb{Q}$ hence we conclude that $\mathbb{Q}[\omega + \omega^{-1}]$ and degree $\frac{\phi(m)}{2}$ over $\mathbb{Q}$.
Let $g$ be the number of prime factors in $\mathbb{Q}[\omega + \omega^{-1}]$, then by another standard theorem $fg = \frac{\phi(m)}{2}$ and we get $g = \frac{\phi(m)}{2f}$.
Thus, if $K=\mathbb{Q}[\omega + \omega^{-1}]$, then
$$ p\mathcal{O}_K = \prod_{\ell = 1}^{\frac{\phi(m)}{2f}} \mathfrak{P}_{\ell}$$
where $f$ is the smallest positive integer satisfying $p^f \equiv \pm 1 \pmod m$.
