Combination of ferroelectric layers and semiconductor heterostructures with 2D electron gas may lead to a number of new applications from high-mobility field effect transistors with ferroelectric gates to quantum dots patterned with polarization domains at sub-micron scale. One of the fundamental obstacles impeding implementation of such devices is associated with the unfavorable relation between the dielectric constants of the classical oxide ferroelectrics and semiconductors. In the present work we demonstrate a way to alleviate this problem by using a gate material that is ferroelectric and semiconducting simultaneously. First, ferroelectricity in semiconductor CdZnTe films is proven and studied using modified piezo-force scanning probe microscopy. Then a rewritable field effect device is demonstrated by local poling of the CdZnTe layer of a CdZnTe/CdTe quantum well, provoking a reversible, nonvolatile change in the resistance of the 2D electron gas.