New Methods for (φ,γ)-modules

Let p be a prime number. The subject of p-adic Hodge theory concerns the interplay between different objects arising from the cohomology of algebraic varieties over p-adic fields. A good introduction to the subject can be found in the notes of Brinon and Conrad [3]; here, we limit discussion to two proofs which we feel are simpler than their counterparts in the literature. Our original motivation for obtaining such proofs was to facilitate generalization of the existing results; such generalizations to the context of relative p-adic Hodge theory can be found in our joint work with Liu [11, 12], and in closely related work of Scholze [13, 14]. Our first topic is the relationship between Galois theory in characteristic 0 and characteristic p provided by a theorem of Fontaine and Wintenberger [7].