Mathematical Areas

Local Fields

Magma supports p-adic fields and their extensions to arbitrary precision. Algorithms include polynomial factorisation over local fields, ramification theory, and local-global compatibility for arithmetic problems.

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Contributors

Local Arithmetic Fields

A package written by Xavier Caruso and David Lubicz (both Rennes) for computing with power series rings over p-adic rings was included in Magma V2.20 (2013) after additional work by Mark Watkins.

Sebastian Pauli (TU Berlin) has made available the Magma implementation of his algorithm for factoring polynomials over local fields. It is also used for factoring ideals and for computing completions of global fields.

The module for lazy power series is based on the ideas of Josef Schicho (Linz).

Handbook Chapter Authors

  • Power, Laurent and Puiseux Series — A. Steel
  • Lazy Power Series Rings — N. Sutherland
  • Algebraic Power Series Rings — T. Beck, M. Harrison
  • Valuation Rings — W. Bosma
  • Galois Rings — A. Steel
  • Newton Polygons — G. Brown, N. Sutherland, SERIES RINGS OVER p-ADIC RINGS, M. Watkins
  • Local Galois Representations — T. Dokchitser