The third homology of the special linear group of a field

We prove that for any infinite field F, the map H3(SLn(F),Z)→H3(SLn+1(F),Z) is an isomorphism for all n≥3. When n=2 the cokernel of this map is naturally isomorphic to 2⋅K3M(F), where KnM(F) is the nth Milnor K-group of F. We deduce that the natural homomorphism from H3(SL2(F),Z) to the indecomposable K3 of F, K3(F)ind, is surjective for any infinite field F.