We now step away from differential modules for a little while, to study the related subject of difference algebra. This is the theory of algebraic structures enriched not with a derivation but with an endomorphism of rings. Our treatment of difference algebra will run largely in parallel with what we did for differential algebra, but in a somewhat abridged fashion; our goal is to say just enough to be able to use difference algebra to say nontrivial things about p-adic differential equations. We will begin to do that in Part V. In this chapter, we introduce the formalism of difference rings, fields, and modules, and the associated notion of twisted polynomials. We then study briefly the analogue of algebraic closure for a difference field. Finally, we make a detailed study of difference modules over a complete nonarchimedean field, culminating with a classification of difference modules for the Frobenius automorphism of a complete unramified p-adic field with algebraically closed residue field (the Dieudonné–Manin classification).