I am reading chapter 9 in Cassels-Frohlich Algebraic Number Theory, this chapter is about class field tower.
In this chapter the following claim is due to Brumer: There exist a function $c(n)$ \begin{gather*}d^p Cl_k\geq t_k^p-c(n)\end{gather*} where $t_k^p$ is the number of rational primes of ramified primes $q$ such that $p$ divide $e_k(q)$ and $n$ is the degree of $k$.
Remark: when $k$ is not Galois then one take $e_k(q)$ to be $gcd_{\mathfrak{q}|q}e(\mathfrak{q}|q)$, feel free to ignore this and assume $k$ is Galois.
When I looked at Brumer's paper (Ramification and class tower of number fields) I couldn't find this claim, I found a similar claim: Let $k$ be a Galois number field of degree $n$, and $r$ the number of generators of its class ideal group and supposed $s$ primes are ramified in $k$ then \begin{gather*}\frac{s}{w(n)}-2n\leq r\end{gather*} where $w(n)$ is the number of distinct prime factors of $n$.
I can't understand how these two claims are related, I don't see how the second inequality implies the first and less important the Galois assumption.
Thank you all!