Kronecker-Weber, wild step for odd $p$: an abelian field of degree $p^k$ unramified outside $p$ lies in $\mathbb{Q}(\zeta_{p^N})$

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Kronecker-Weber, wild step for odd $p$: an abelian field of degree $p^k$ unramified outside $p$ lies in $\mathbb{Q}(\zeta_{p^N})$ · Prove2Me