Let $\mathbf{Q}^{\mathrm{ab}}$ be the maximal abelian extension of the field of rational numbers $\mathbf{Q}$. I'm interested in the following question:


Is it true that $K^{M}\_{2}(\mathbf{Q}^{\mathrm{ab}})/pK^{M}\_{2}(\mathbf{Q}^{\mathrm{ab}})=0$ for any prime $p$ ?


where $K^{M}\_{\ast}$ is Milnor $K$-theory.



The second Milnor $K$-theory of a field
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).


