= Solution
For a <principal unit> $u$, apply the <Hensel lemma> to $F(T)=T^{p-1}-u$. The residue class one is a root, and $F'(1)=p-1$ is a <unit>. Thus there is a unique $v\in1+\pi_K\mathcal O_K$ with
$$
\boxed{v^{p-1}=u.}
$$
In particular it is a <unit> of $\mathcal O_K$. Choose this $v$ for the $u$ just obtained and put $\alpha=\pi_K/v$. Then $\alpha^{p-1}=-p$, so $\mathbb Q_p(\alpha)\subseteq K$. The polynomial $T^{p-1}+p$ is <Eisenstein>, giving degree $p-1$ for the left-hand field. The <p-adic cyclotomic extension> $K$ also has degree $p-1$, so
$$
\boxed{K=\mathbb Q_p\!\left(\sqrt[p-1]{-p}\right).}
$$
Back to article page
