This work presents a polynomial version of the well-known extended Kalman filter (EKF) for the state estimation of nonlinear discrete-time stochastic systems. The proposed filter, denoted polynomial EKF (PEKF), consists in the application of the optimal polynomial filter of a chosen degree mu to the Carleman approximation of a nonlinear system. When mu = 1 the PEKF algorithm coincides with the standard EKE, For the filter implementation the moments of the state and output noises up to order 211 are required. Numerical simulations compare the performances of the PEKF with those of some other existing filters, showing significant improvements.