= Ramification groups of the splitting field of Xp minus p over the p-adic numbers
{title2=$\mathbb Q_p(\zeta_p,\sqrt[p]{p})/\mathbb Q_p$}
For odd $p$, the splitting field $K=\mathbb Q_p(\zeta_p,\sqrt[p]{p})$ is totally ramified of degree $p(p-1)$. With
$$
\varpi=\frac{\zeta_p-1}{\sqrt[p]{p}},
$$
the lower ramification groups are
$$
G_0=\operatorname{Gal}(K/\mathbb Q_p),\qquad
G_i\cong\mathbb Z/p\mathbb Z\ (1\leq i\leq p),\qquad
G_i=1\ (i\geq p+1).
$$
To verify the filtration, write $\alpha^p=p$ and $\lambda=\zeta_p-1$. The <Eisenstein polynomials> for $\alpha$ and $\lambda$ give coprime degrees $p$ and $p-1$, so their compositum has degree $p(p-1)$. Its element $\varpi=\lambda/\alpha$ has $p$-adic <valuation> $1/[p(p-1)]$, forcing total ramification. For an automorphism $\sigma_{a,b}$ with $\alpha\mapsto\zeta_p^a\alpha$ and $\zeta_p\mapsto\zeta_p^b$,
$$
\frac{\sigma_{a,b}(\varpi)}{\varpi}=\zeta_p^{-a}\frac{\zeta_p^b-1}{\zeta_p-1}.
$$
Its residue is $b$, so the normalized displacement <valuation> is one if $b\ne1$. If $b=1$ and $a\ne0$, that valuation is $1+v_K(\zeta_p^{-a}-1)=p+1$. The <uniformizer criterion for lower ramification groups> now gives every group displayed above.
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