We formulate the MFG limit for N interacting agents with a common noise as a single quasi-linear deterministic infinite-dimensional partial differential second order backward equation. We prove that any (regular enough) solution provides an (Formula presented.) -Nash-equilibrium profile for the initial N-player game. We use the method of stochastic characteristics to provide the link with the basic models of MFG with a major player. We develop two auxiliary theories of independent interest: sensitivity and regularity analysis for McKean-Vlasov SPDEs and the (Formula presented.) -convergence rate for the propagation of chaos property of interacting diffusions.