Let $\mathbb{Q(\zeta)}/\mathbb{Q}$ be a galois extension of degree $p-1$ where $\zeta=e^{\frac{2 \pi i}{p}} $
Let $\mathfrak{b}$ be an ideal above $q$ Where q is a prime in $\mathbb{Z}$ different from $p$
Then We have $<q> =\mathfrak{b}_{1}^{e}......\mathfrak{b}_{r}^{e} $ , by ramification theory we have $rfe=p-1$ where $f $ the is inertial degree and $e$ is the ramification index .
We have
$\mathcal{N}(\mathfrak{b})=|\mathcal{O}/\mathfrak{b}|=|\mathbb{F}_{q}|^{f}=q^{f}$
Where $\mathcal{O}=\mathbb{Z}[\zeta]$ is the ring of integers of $\mathbb{Q(\zeta)}/\mathbb{Q}$
By theorem 9.5 in the book of algebraic number and Fermat last Theorem by Stewart and D.Tall, there is $\alpha \in \mathfrak{b} $ s.t
$$ \mathcal{N}(\alpha) \leq (\frac{ \pi }{2})^{t} \sqrt{\Delta}q^{f} $$ Thus we get
$\frac{ \log q |\mathcal{N}(\alpha)| \pi^{t}}{2^{t} \sqrt{\Delta}} \leq f $
Where $\Delta$ is the discriminant of $\mathbb{Q(\zeta)}/\mathbb{Q}$ and $t $ is the number of complex embedding in $\mathbb{C} $
I am asking whether the above inequality for $f $ is true ? and is there a better lower bound for $f$?