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Theorem: Let $\zeta_m$ be a primitive mth root of unity, and $K$ a field. If $\zeta_m \in K$, then every $\mathbb{Z}/m\mathbb{Z}$-extension of $K$ is of the form $K(\alpha^\frac{1}{m})$ for some $\alpha \in K^*$ with the property that $\alpha^\frac{1}{d} \not\in K$ for any proper divisor $d$ of $m$.

Proof: Let $\alpha \in K^*$ be such that $\alpha^\frac{1}{d} \not\in K$ for all proper divisors $d$ of $m$. Every automorphism of $K(\alpha^\frac{1}{m})$ over $K$ has the form $\alpha^\frac{1}{m} \mapsto \zeta_m^k\alpha^\frac{1}{m}$ for $k \in \mathbb{Z}/m\mathbb{Z}$; this defines a group homomorphism $$\text{gal}(K(\alpha^\frac{1}{m})/K) \to \mathbb{Z}/m\mathbb{Z}.$$ This map is injective since an automorphism is uniquely determined by its action on $\alpha^\frac{1}{m}$. If $m$ is prime, then its clear the above is also a surjection, since the image of the Galois group in $\mathbb{Z}/m\mathbb{Z}$ would be a non-trivial subgroup of a cyclic group of prime order i.e. the whole group. That the image of the Galois group is a non-trivial subgroup follows from the hypothesis that $\alpha^\frac{1}{m} \not\in K$.

In the case that $m$ is not prime, note the definition of the map $$\text{gal}(K(\alpha^\frac{1}{m})/K) \to \mathbb{Z}/m\mathbb{Z}$$ is compatible with replacing $m$ with $p$, where $p$ is a prime factor of $m$. This I follow. The author then claims that it follows from this that the image of $\text{gal}(K(\alpha^\frac{1}{m})/K)$ in $\mathbb{Z}/m\mathbb{Z}$ cannot be contained in $p\mathbb{Z}/m\mathbb{Z}$ for any prime divisor $p$ of $m$. I don't follow why this is true. Assuming it is, it is clear to me why this concludes the proof of surjectivity in the non-prime case, since the subgroups $p\mathbb{Z}/m\mathbb{Z}$ are maximal in $\mathbb{Z}/m\mathbb{Z}$ and so if the image of the Galois group is contained in none, it must be the whole group.

Help understanding the parts I've highlighted as not following would be very useful.

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This is theorem 25.16 of the Algebra III notes by Stevenhagen, which may be found here: https://websites.math.leidenuniv.nl/algebra/.

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