= Solution
Let $\alpha=p^{1/p}$ and let $\zeta$ be a primitive $p$th <root of unity>. The roots are $\alpha\zeta^j$, so $L=\mathbb Q(\alpha,\zeta)$. Eisenstein gives $[\mathbb Q(\alpha):\mathbb Q]=p$, while the prime cyclotomic extension has degree $[\mathbb Q(\zeta):\mathbb Q]=p-1$. Both degrees divide $[L:\mathbb Q]$, and they are coprime. Conversely adjoining $\alpha$ to the <cyclotomic field> has degree at most $p$. These two bounds give
$$
\boxed{[L:\mathbb Q]=p(p-1).}
$$
Restriction to $E=\mathbb Q(\zeta)$ gives a <normal subgroup> $N=\operatorname{Gal}(L/E)$ of order $p$, hence cyclic, with quotient $\operatorname{Gal}(E/\mathbb Q)\cong C_{p-1}$. Restriction is onto because embeddings of the intermediate <field> extend in the finite Galois <splitting field>. Thus $1\trianglelefteq N\trianglelefteq G$ has abelian factors, proving $\boxed{G\text{ is solvable}}$. For $p=2$ the <cyclotomic field> is just $\mathbb Q$, and the same argument gives degree two and a <cyclic group>.
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