By the Milnor-Bloch-Kato conjecture, this is equivalent to $\mathrm{Br}(\mathbf{Q}^{\mathrm{ab}})[p] = 0$ (by the Kummer sequence $1 \to \mu_p \to \mathbf{G}_m \to \mathbf{G}_m \to 1$, Hilbert 90 $H^1(K,\mathbf{G}_m) = 0$ and $H^2(K,\mathbf{G}_m) = \mathrm{Br}(K)$). This follows from [Neukirch-Schmidt-Wingberg, Cohomology of Number Fields https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/NSW2.2.pdf ], Proposition (8.1.14) (ii).
