Kronecker-Weber, wild step for $p = 2$: an abelian field of degree $2^k$ unramified outside $2$ lies in $\mathbb{Q}(\zeta_{2^N})$

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Kronecker-Weber, wild step for $p = 2$: an abelian field of degree $2^k$ unramified outside $2$ lies in $\mathbb{Q}(\zeta_{2^N})$ · Prove2Me