Abstract
It is well known that for certain fields K of characteristic p /= 0 there exists finite extensions E/K that are not separable, yet contain no purely inseparable elements. However, the question of when such a phenomenon occurs has not been widely recorded in the literature. Certain of the well known books give only examples, (2, p. 136, Exercise 17) and (5, p. 49, Exercise 3). It seems natural to ask for more general information, and it will be the object of this thesis to make some remarks in this direction. | We write E/K to Indicate that E is a finite-dimensional extension field of K. The letter p is reserved for the characteristic of the field under discussion, and since our questions generally evaporate when p = 0, we shall always assume that p > 0. | Barnes (1) will be used as a basis of terminology, definitions not normally found in elementary algebra courses will be given.