Nanoscale corundum powders are obtained by first producing an Al₂O₃ precursor by adding seed crystals to an aqueous solution of an aluminium compound and adding a base and then converting the Al₂O₃ precursor into corundum by calcination at a high temperature. Before the calcination, the salts that are present in addition to the Al₂O₃ precursor are separated off. The resulting product is calcined at temperatures of 700 to 975 C. and any fines that may be present are removed. The resulting corundum powders can be sintered at temperatures of 1200 C. to produce compacts or components of multiple layer systems.