= Solution
Over $K=\mathbb Q_p$, consider the two <Lubin–Tate series>
$$
f(X)=(1+X)^p-1,
\qquad
g(X)=X^p+pX.
$$
Both have linear term $pX$ and reduce to $X^p$ modulo $p$. The nonzero roots of $f$ are $\zeta_p^i-1$, while the nonzero roots of $g$ satisfy
$$
X^{p-1}=-p.
$$
The <Lubin–Tate change of series> $h:\mathcal F_f\to\mathcal F_g$ and its inverse have coefficients in $\mathbb Z_p$ and converge on the maximal ideal. They therefore give mutually inverse bijections between the first torsion sets without changing the fields generated by them. Hence the <first Lubin–Tate torsion fields for the p-adic numbers> satisfy
$$
\mathbb Q_p(\zeta_p-1)
=\mathbb Q_p\left(\sqrt[p-1]{-p}\right).
$$
Since $\mathbb Q_p(\zeta_p-1)=\mathbb Q_p(\zeta_p)$, the required equality follows.
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