of algebraic fields, that is, fields in the ordinary sense, onily if the characteristic p exceed 0. In this case an extension which is monadic in the sense just described may be produced by adjoining a p-th root not already in the field. Our purpose in this paper is to present in organized form the still rudimentary theory of these phenomena, and to point out their decisive importance in the algebraic theory of difference equations. They seem, in fact, to mark a point byond which one can no longer use the theory of polynomial ideals or the algebraic theory of differential equations as a guide to the study of difference equations, but must expect phenomena which are sti generis. Our results also suggest interesting questions concerning field structure. It would, for example, be desirable to characterize those fields which possess incompatible or monadic extensions. Our principal result is a first step toward such a characterization; fields which are algebraically closed have no finite extensions of these types.