A nonvolatile memory element and associated production methods and memory element arrangements are presented. The nonvolatile memory element has a changeover material and a first and second electrically conductive electrode present at the changeover material. To reduce a forming voltage, a first electrode has a field amplifier structure for amplifying a field strength of an electric field generated by a second electrode in a changeover material. The field amplifier structure is a projection of the electrodes which projects into the changeover material. The memory element arrangement has multiple nonvolatile memory elements which are arranged in matrix form and can be addressed via bit lines arranged in column form and word lines arranged in row form.