Neuromorphic computing architectures enable the dense colocation of memory and processing elements within a single circuit. This colocation removes the communication bottleneck of transferring data between separate memory and computing units as in standard von Neuman architectures for data-critical applications including machine learning. The essential building blocks of neuromorphic systems are nonvolatile synaptic elements such as memristors. Key memristor properties include a suitable nonvolatile resistance range, continuous linear resistance modulation, and symmetric switching. In this work, we demonstrate voltage-controlled, symmetric and analog potentiation and depression of a ferroelectric Hf0.57Zr0.43O2 (HZO) field-effect transistor (FeFET) with good linearity. Our FeFET operates with low writing energy (fJ) and fast programming time (40 ns). Retention measurements have been performed over 4 bit depth with low noise (1%) in the tungsten oxide (WOx) readout channel. By adjusting the channel thickness from 15 to 8 nm, the on/off ratio of the FeFET can be engineered from 1 to 200% with an on-resistance ideally >100 kω, depending on the channel geometry. The device concept is using earth-abundant materials and is compatible with a back end of line (BEOL) integration into complementary metal-oxide-semiconductor (CMOS) processes. It has therefore a great potential for the fabrication of high-density, large-scale integrated arrays of artificial analog synapses.