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 m to the Carleman approximation of a nonlinear system. When m=1 the PEKF algorithm coincides with the standard EKF. For the filter implementation the moments of the state and output noises up to order 2 are required. Numerical simulations compare the performances of the PEKF with those of some other existing filters, showing significant improvements.