When an erase voltage is applied to the sources of data erasable and rewritable memory cells each having a floating gate, the erasure characteristics of the memory cells can be improved by controlling the rise time of the erase voltage or by increasing the erase voltage stepwise. In test mode, no row lines are selected by a row decoder and further the sources of the respective memory cells are set to ground level. Under these conditions, in case there exists an overerased memory cell, this cell is turned on due to depletion, so that it is possible to detect the presence of the overerased memory cell on the basis of change in potential of the column line connected to this turned on memory cell. A differential amplifier is used to detect the change in potential of the column line. In the test mode, the potential of the column lines is compared with a reference potential applied to a dummy column line, and a source bias generating circuit applies a test potential suitable for test to the respective sources of the cells, to shift the threshold level of the respective cells in a positive direction, for instance. By applying this test potential to the cells, it is possible to detect the pseudo-threshold level shifted in the positive direction; that is, to detect the overerased status of the memory cell more properly. Further, the erasure is effected until the threshold level of a memory cell of the highest erasure speed reaches a predetermined level, irrespective of the threshold distribution width of the memory cells, thus realizing a higher speed access to the device of narrower threshold distribution width, as compared with the conventional device.