Lemmermeyer's proof of Kronecker-Weber

Question

I am reading Lemmermeyer's proof of Kronecker-Weber, specifically the proof of Proposition 1.1. The paper is linked here: https://arxiv.org/pdf/1108.5671.pdf

The setup is as follows: $K/\mathbb{Q}$ is a cyclic extension of degree $p$, $F=\mathbb{Q}(\zeta_{p})$, and $L=KF$ is the compositum. Now, $L/F$ is a cyclic extension, so we can write it in the form $F(\mu^{1/p})$. Following the proof of Lemma 2.1, he concludes that $(1-\zeta)$ cannot divide $\mu$. I am unable to see why this is true. Similarly, in the next line, he again says that if $\mathfrak{p}$ is a prime ideal not dividing $p$, then $\mathfrak{p}^{bp}||\mu$ for some integer $b$. I cannot see why this its true either, because the previous proposition does not imply that this happens for primes of the form which are $1 \pmod{p}$, as these split completely. If I could get a hint on how to proceed with respect to these doubts, I would be thankful.

Edit: I have figured out why the second point holds. This can be seen by the fact that $L/F$ is unramified outside $p$, and by looking at the prime factorization of $(\mu^{1/p})$ in $O_{L}$. However, I am still dumbfounded as to why the first point holds.