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Oscar.jl
  • Welcome to OSCAR
    • Architecture
    • Notes for users of other computer algebra systems
    • Frequently Asked Questions
    • Saving and loading files
    • 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
      • Introduction
      • Elements
      • Morphisms
      • Structural Computations
    • Group characters
    • Group recognition
    • Tables of Marks
    • 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
    • Algebraic numbers
    • Algebraic closure of finite prime 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
      • Vinberg's algorithm
    • Introduction
      • Contact
      • 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
      • Ideals and Quotient Rings as Modules
      • 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 Tori
    • Invariants of Linearly Reductive Groups
    • Introduction
      • General schemes
      • Affine schemes
      • Morphisms of affine schemes
      • Rational Points on Affine Schemes
      • Covered schemes
      • Coverings
      • Morphisms of covered schemes
      • Projective schemes
      • Morphisms of projective schemes
      • Rational Points on Projective Schemes
      • Sheaves on covered schemes
      • Cycles and divisors
      • Affine Algebraic Sets
      • Projective Algebraic Sets
      • Affine Varieties
      • Projective Varieties
      • Affine plane curves
      • Projective Curves
      • Projective Plane Curves
      • Rational Parametrizations of Rational Plane Curves
      • Automorphism Groups of K3 surfaces
      • Adjunction Process for Surfaces
      • Rational Parametrization of Rational Surfaces
      • Nongeneral Type Surfaces in $\mathbb P^4$
      • Classifier/identifier specifically for du Val singularities
      • Sheaves on Projective Space
      • 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
      • Toric Schemes
      • Toric Ideal Sheaves (Experimental)
      • Toric Blowups (Experimental)
      • Some Special Ideals
      • Architecture of affine schemes
    • Introduction
    • Tropical semirings, matrices, and polynomials
    • Tropical semiring maps
    • Tropical varieties
    • Tropical hypersurfaces
    • Tropical curves
    • Tropical linear spaces
    • Groebner theory
    • Tropicalization of polynomial ideals
    • Introduction
      • Introduction
      • Creating PBW-Algebras
      • Ideals in PBW-algebras
      • GR-Algebras: Quotients of PBW-Algebras
    • Free Associative Algebras
    • Graphs
    • Matroids
    • Simplicial Complexes
    • Phylogenetic Trees
      • Partitions
      • Tableaux
      • Schur polynomials
      • Compositions
    • Introduction
    • GAP's SLPs
    • AbstractAlgebra's polynomial interface
  • References
  • Index
    • Introduction for new developers
    • Developer Style Guide
    • Documenting OSCAR code
    • Printing in OSCAR
    • Debugging OSCAR Code
    • Caching parent objects in OSCAR
    • Serialization
    • Design Decisions
    • GAP Integration
    • Release management
      • AbstractCollection
      • SubObjectIterator
    • Adding new projects to experimental
      • Introduction
      • Graphical Models
      • Gaussian Graphical Models
      • Conditional independence statements
      • Discrete random variables
      • Algebraic Phylogenetics
      • Double complexes – the user's interface
      • Advice for the programmer
      • Usage
      • Special ideals used for benchmarking
      • Introduction
      • Construction and basic functionality
      • Cox rings
      • Matroid Realization Spaces
      • Introduction
      • Access to precomputed OD data
      • Criteria for computing orthogonal discriminants
      • Miscellaneous functions
      • Quadratic forms and isometries
      • Quadratic spaces with isometry
      • Lattices with isometry
      • Enumeration of isometries
      • Nikulin's theory on primitive embeddings
      • -
      • Introduction
      • Lie algebras
      • Ideals and Lie subalgebras
      • Lie algebra homomorphisms
      • Lie algebra modules
      • Lie algebra module homomorphisms
      • Cartan Matrices
      • Introduction
      • Functions for a monomial basis of highest weight modules
      • Partitioned Permutations
      • Welcome to FTheoryTools
      • Functionality for all F-theory models
      • Weierstrass models
      • Global Tate models
      • Hypersurface models
      • Literature constructions
      • G4-Fluxes
      • Introduction
      • Abstract Varieties
      • Abstract Bundles
      • Abstract Variety Maps
      • Blowups
      • Schubert Calculus
      • Bott Formulas
      • Illustrating Examples From Enumerative Geometry
Version
  • Number Theory
  • Introduction
  • Introduction
GitHub

Introduction

The number theory part of OSCAR provides functionality for algebraic number theory.

It is under development with regard to providing both the functionality and the documentation.

General textbooks offering details on theory and algorithms include:

  • [Coh93]
  • [Coh00]
  • [Mar18]
  • [PZ97]

Contact

Please direct questions about this part of OSCAR to the following people:

  • Claus Fieker,
  • Tommy Hofmann.

You can ask questions in the OSCAR Slack.

Alternatively, you can raise an issue on github.

« Vinberg's algorithmIntroduction »

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