Cyclotomic local different exponent
= Cyclotomic local different exponent
{title2=$p^{r-1}(r(p-1)-1)$}
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 $p^{r-1}$.