 Cyclotomic local different exponent

For $K_r={\mathbb{Q}}_p({\zeta }_{p^r})$, the uniformizer ${\zeta }_{p^r}-1$ has different exponent $p^{r-1}(r(p-1)-1)$. Differentiate the cyclotomic polynomial at ${\zeta }_{p^r}$: the numerator contributes $r{p}^{r-1}(p-1)$ to the valuation, while the denominator ${\zeta }_p-1$ contributes $p^{r-1}$.