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Oscar.jl
  • Welcome to OSCAR
    • Architecture
    • Notes for users of other computer algebra systems
    • Frequently Asked Questions
    • Serialization
    • Complex Algorithms in OSCAR
    • Introduction
    • Basics
    • Subgroups
    • Quotients
    • Products of groups
    • Permutation groups
    • Finitely presented groups
    • Polycyclic groups
    • Matrix groups
    • Group Actions
    • Group homomorphisms
    • Groups of automorphisms
    • Group libraries
    • Abelian Groups
    • Group characters
    • Introduction
    • Ring functionality
    • Integers
      • Univariate polynomial functionality
      • Sparse distributed multivariate polynomials
      • Power series
      • Generic Puiseux series
      • Power series and Laurent series
      • Puiseux series
    • Introduction
    • Field functionality
    • Rationals
    • Factored Elements
    • Class Field Theory
    • Generic fraction fields
      • Padics
      • Qadics
    • Finite fields
    • Introduction
    • Sparse linear algebra
    • Matrix functionality
    • Generic matrix algebras
      • Finitely presented modules
      • Free Modules and Vector Spaces
      • Submodules
      • Quotient modules
      • Direct Sums
      • Module Homomorphisms
      • Introduction
      • Spaces
      • Lattices
      • Genera for hermitian lattices
      • Integer Lattices
      • Genera of Integer Lattices
      • Discriminant Groups
    • Introduction
      • Introduction
      • Number field operations
      • Element operations
      • Internals
      • Introduction
      • Orders
      • Elements
      • Ideals
      • Fractional ideals
    • Abelian closure of the rationals
    • Galois Theory
    • Introduction
      • Introduction
      • Constructions
      • Polyhedron and polymake's Polytope
      • Auxiliary functions
    • Cones
    • Polyhedral Fans
    • Polyhedral Complexes
    • Linear Programs
    • Mixed Integer Linear Programs
    • Subdivisions of Points
    • Introduction
    • Creating Multivariate Rings
    • Ideals in Multivariate Rings
    • Affine Algebras and Their Ideals
    • Localized Rings and Their Ideals
      • Introduction
      • Free Modules
      • Subquotients
      • Operations on Modules
      • Operations on Module Maps
      • Chain and Cochain Complexes
      • Homological Algebra
      • Monomial Orderings
      • Gröbner/Standard Bases Over Fields
      • Gröbner/Standard Bases Over $\mathbb Z$
      • Binomial Primary Decomposition
      • A Framework for Localizing Rings
      • Localizations of modules over computable rings
    • Introduction
    • Invariants of Finite Groups
    • Invariants of Linearly Reductive Groups
    • Introduction
      • General schemes
      • Affine schemes
      • Morphisms of affine schemes
      • Architecture of affine schemes
      • Covered schemes
      • Coverings
      • Introduction
      • Normal Toric Varieties
      • Cyclic Quotient Singularities
      • Toric Divisors
      • Toric Divisor Classes
      • Toric Line Bundles
      • Line bundle cohomology with cohomCalg
      • Cohomology Classes
      • Subvarieties
      • The Chow ring
      • ToricMorphisms
      • Introduction
      • Affine Toric Schemes
      • Normal Toric Schemes
      • Introduction
      • Curves
      • Rational Parametrizations of Rational Plane Curves
      • Standard Constructions in Algebraic Geometry
      • Algebraic Surfaces
    • Introduction
      • Introduction
      • Creating PBW-Algebras
      • Ideals in PBW-algebras
      • GR-Algebras: Quotients of PBW-Algebras
    • Free Associative Algebras
    • Graphs
    • Matroids
    • Simplicial Complexes
    • Introduction
    • GAP's SLPs
    • AbstractAlgebra's polynomial interface
  • References
  • Index
    • Introduction for new developers
    • Developer Style Guide
    • Design Decisions
    • Documenting OSCAR code
    • Debugging OSCAR Code
    • Serialization
      • AbstractCollection
      • SubObjectIterator
Version
  • Fields
  • Introduction
  • Introduction
Edit on GitHub
  • Introduction

Introduction

The fields part of OSCAR provides functionality for handling various kinds of fields:

  • the field of rationals
  • Number fields
  • Generic fraction fields
  • local fields (Padics and Qadics)
  • finite fields

General textbooks offering details on theory and algorithms include:

  • Henri Cohen (1993)
  • Henri Cohen (2000)
  • Rudolf Lidl, Harald Niederreiter (1997)
  • Daniel A. Marcus (2018)
  • M. Pohst, H. Zassenhaus (1997)
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