In this paper, we consider mean field games in the presence of common noise relaxing the usual independence assumption of individual random noise. We assume a simple linear model with terminal cost satisfying a convexity and a weak monotonicity property. Our main result is showing existence and uniqueness of a mean field game solution using the stochastic maximum principle. The uniqueness is a result of a monotonicity property similar to that of Lasry and Lions. We use the Banach fixed point theorem to establish an existence over a small time duration and show that it can be extended over an arbitrary finite time duration.