Fast arithmetic in unramified p-adic fields

Abstract

Let p be prime and Zpn the degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast algorithms for common operations in Zpn modulo p^N. 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, Teichmuller lifting and computing minimal polynomials.