$\newcommand{\ab}{\mathrm{ab}}$Let $\mathbb{Q}^{\ab}$ denote the maximal abelian extension of $\mathbb{Q}$. I have heard the absolute Galois group of $\mathbb{Q}^{\ab}$ is projective (e.g. see this page on the Shafarevich conjecture) but am having trouble finding a source for this fact.
The above page suggests as reference Cohomologie Galoisienne by Serre, and I believe Kronecker-Weber + Serre's Chapt. 2 Prop. 9 shows the cohomological dimension of $G_{\mathbb{Q}^{\ab}}$ is $\leq 1$, which is equivalent to projectivity. Is there another (self-contained) way to show this fact?
