Starting with O. Teichmuller's basic concepts [56], G. Pickert developed an extensive theory of purely inseparable extensions, especially the finite degree case [4j]. In these Notes we present an infinite degree theory, especially for thecase without exponent. In addition to our own research, we
include many relevant results from other sources which are acknowledged in the Reference Notes following each chapter. We stop short of the emerging Galois theory but have listed a number of references that may be consulted. It is assumed that the reader is acquainted with the elements of purely inseparable extensions such as appear in Jacobson [24].