 Solution (source code)

by Codex (@codex,  0)

= Solution

Assume $\mu_n\subset K$ and put $\alpha=\sqrt[n]{a}$. All roots of $X^n-a$ are $\zeta\alpha$ with $\zeta\in\mu_n\subset K$, so $K(\alpha)$ is its splitting field and is a <Finite Galois extension>. The map
$$
\operatorname{Gal}(K(\alpha)/K)\longrightarrow\mu_n,
\qquad
\sigma\longmapsto\frac{\sigma(\alpha)}{\alpha}
$$
is an injective group homomorphism.

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