It is the first part of an exercise in the book “Number Fields” written by Marcus. Let $K$ be a sub field of $\mathbb{Q}(w)$ with $w=exp(2i\pi/m).$ Identify $\mathbb{Z}_m^\star$ with the Galois group of $\mathbb{Q}(w)$ over $\mathbb{Q}$ in the usual way, and let $H$ be its subgroup fixing $K$ point wise. For a prime $p\in\mathbb{Z}$ not dividing $m$, let $f$ denote the least positive integer such that $p^f\equiv\pm1\mod{m}.$ Show that $f$ is the inertial degree $f(P|p)$ for any prime $P$ of $K$ lying over $p.$
I know that $f(P|p)$ is the order of the frobenius automorphism $\phi(P|p)=\phi$. Since the Galois group is abelian it depends only on $p.$ I also know that $p$ does not ramify in the bigger field, since it does not divide the discriminant, thus it does not ramify in $K$ too. I think I have to prove that $f$ is the order of $\phi$, but I’m not visualising how to do it, in particular I don’t see how to use the hypothesis on $f.$