Why Galois group of $K[ζ_{p^r}]/K[ζ_p]$ is cyclic? Why Galois group of $K[ζ_{p^r}]/K[ζ_p]$,p is a odd prime,is cyclic?
Why Galois group of $K[ζ_{2^r}]/K[ζ_4]$ is cyclic?
ζ denotes a primitive root of unity.K is a number field.
These facts are used in the proof of “local n-th powers for all but finitely many primes are indeed global powers when $K[ζ_{2^{ord_2(n)}]/K$ is cyclic” in Milne’s CFT Chapter VIII theorem 1.4,I’m almost sure he didn’t prove these facts before in FT,ANT or CFT.
For his notes: https://www.jmilne.org/math/CourseNotes/cft.html

In Chapter 6 of his ANT notes, Milne shows that $Gal(\mathbb{Q}[\zeta_{p^{r}}]/\mathbb{Q})\simeq(\mathbb{Z}/p^{r}\mathbb{Z})^{\times}$. In
particular, $Gal(\mathbb{Q}[\zeta]/\mathbb{Q}{})\simeq(\mathbb{Z}/p\mathbb{Z}{})^{\times}$, and so $Gal(\mathbb{Q}[\zeta_{p^{r}}%
]/\mathbb{Q}[\zeta]\mathbb{)}$ is the kernel of $(\mathbb{Z}/p^{r}\mathbb{Z}{})^{\times}\rightarrow(\mathbb{Z}/p\mathbb{Z}{})^{\times}$. This is cyclic for $p$ odd --- see Chapter 3 of his Group Theory notes. When you replace $\mathbb{Q}$ with $K$, you replace the Galois group with a quotient [Oops, subgroup].
