In Serre's book Galois cohomology he describes an example of a field with trivial Brauer group that is not of dimension $\leq 1$, as follows:
Exercise II.3.1.1. Let $k_0$ be a field of characteristic $0$ such that $k_0$ has no nontrivial abelian extensions, but $k_0 \neq \overline{k_0}$. (An example is the compositum of all finite solvable Galois extensions of $\mathbb{Q}$). Let $k = k_0((T))$, then $\text{Br}(k) = 0$, while $k$ is not of dimension $\leq 1$.
I can understand the second part of the statement, but the first part is giving me some trouble.
The questions I have are as follows:
Does $\text{Br}(k_0) = 0$ hold for any field (with characteristic zero) and no nontrivial abelian extensions?
Can one compare the relative Brauer groups $\text{Br}(\overline{k_0}/k_0)$ and $\text{Br}(\overline{k_0}((T))/k_0((T)))$? Are there conditions for which they are equal?
The reason for these questions is as follows (using $k = k_0((T))$ for readability): from the tower $\overline{k} / \overline{k_0}((T)) / k$, denoting $G = \text{Gal}(\overline{k}/k)$, $N = \text{Gal}(\overline{k}/\overline{k_0}((T)))$ and $H = \text{Gal}(\overline{k_0}((T))/k)$ one obtains an exact sequence $1 \to N \to G \to H \to 1$, which results in an exact sequence
$0 \to \text{Br}(\overline{k_0}((T))/k) \to \text{Br}(k) \to \text{Br}(\overline{k_0}((T))$.
However, this last Brauer group is zero as the field is $C_1$. Thus it suffices to show that $\text{Br}(\overline{k_0}((T))/k) = 0$. This is where the second (and subsequently the first) question comes in.
I would appreciate any hints and/or references.