We present a study of the experimental determination of the forward-backward asymmetry in the process $e^+e^-\to t\bar t$ and in the subsequent $t\to Wb$ decay, studied in the context of the International Linear Collider. This process probes the elementary couplings of the top quark to the photon, the $Z$ and the $W$ bosons at a level of precision that is difficult to achieve at hadron colliders. Measurement of the forward-backward asymmetry requires excellent $b$ quark identification and determination of the quark charge. The study reported here is performed in the most challenging all-hadronic channel $e^+e^- \to b\bar b q\bar q q\bar q$. It includes realistic details of the experimental environment, a full Monte Carlo simulation of the detector, based on the Silicon Detector concept, and realistic event reconstruction. The forward-backward asymmetries are determined to a precision of approximately 1% for each of two choices of beam polarization. We analyze the implications for the determination of the $t\bar t Z$ and $Wtb$ couplings. Show Partial Abstract