By the Milnor-Bloch-Kato conjecture, this is equivalent to $\mathrm{Br}(\mathbf{Q}^{\mathrm{ab}})[p] = 0$. 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).