Fast arithmetic in unramified p-adic fields

Let p be prime and Zpn a degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast deterministic algorithms for common operations in Zpn modulo p . Combining existing methods with recent work of Kedlaya and Umans about modular composition of polynomials, we achieve quasi-linear time algorithms in the parameters n and N , and quasi-linear or quasi-quadratic time in log p, for most basic operations on these fields, including Galois conjugation, Teichmüller lifting and computing minimal polynomials.