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The Magma Handbook
==================

.. image:: /_static/MagmaLogoOfficial.png
   :alt: Magma logo
   :width: 200px
   :align: center
   :class: logo-light

.. image:: /_static/magmadark.png
   :alt: Magma logo
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   :class: logo-dark

Magma is a computer algebra system for computations in algebra, number theory,
algebraic geometry, and combinatorics. This handbook documents the functions,
data types, and algorithms available in the system.

.. grid:: 3
   :gutter: 2
   :class-container: index-quicklinks

   .. grid-item-card:: Preface
      :link: preface
      :link-type: doc
      :text-align: center

      About Magma and the handbook

   .. grid-item-card:: General Index
      :link: genindex
      :link-type: ref
      :text-align: center

      Alphabetical index of all functions

   .. grid-item-card:: References
      :link: references
      :link-type: doc
      :text-align: center

      Bibliography

----

.. toctree::
   :hidden:

   preface
   references
   MagmaLanguage/index-magma-language
   SetsSequencesMappings/index-sets-sequences-mappings
   BasicRings/index-basic-rings
   MatricesLinearAlgebra/index-matrices-linear-algebra
   LatticesQuadraticForms/index-lattices-quadratic-forms
   GlobalFields/index-global-fields
   LocalFields/index-local-fields
   FiniteGroups/index-finite-groups
   FinitelyPresentedGroups/index-finitely-presented-groups
   RepresentationTheory/index-representation-theory
   LieTheory/index-lie-theory
   Modules/index-modules
   Algebras/index-algebras
   CommutativeAlgebra/index-commutative-algebra
   AlgebraicGeometry/index-algebraic-geometry
   ArithmeticGeometry/index-arithmetic-geometry
   ModularArithmeticGeometry/index-modular-arithmetic-geometry
   Topology/index-topology
   Geometry/index-geometry
   Combinatorics/index-combinatorics
   CodingTheory/index-coding-theory
   Cryptography/index-cryptography
   Optimization/index-optimization

.. toctree::
   :hidden:
   :caption: Draft (machine-generated, unreviewed)

   _draft/index

.. grid:: 1 2 3 3
   :gutter: 3
   :class-container: index-topic-grid

   .. grid-item-card:: The Magma Language
      :link: MagmaLanguage/index-magma-language
      :link-type: doc

      Syntax, types, control flow, procedures, functions and the Magma programming model.

   .. grid-item-card:: Sets, Sequences & Mappings
      :link: SetsSequencesMappings/index-sets-sequences-mappings
      :link-type: doc

      Fundamental aggregate types: sets, multisets, sequences, tuples and mappings.

   .. grid-item-card:: Basic Rings
      :link: BasicRings/index-basic-rings
      :link-type: doc

      Integers, rationals, finite fields, residue class rings and polynomial rings.

   .. grid-item-card:: Matrices & Linear Algebra
      :link: MatricesLinearAlgebra/index-matrices-linear-algebra
      :link-type: doc

      Matrix construction, arithmetic, decompositions and solving linear systems.

   .. grid-item-card:: Lattices & Quadratic Forms
      :link: LatticesQuadraticForms/index-lattices-quadratic-forms
      :link-type: doc

      Integral lattices, quadratic forms, reduction algorithms and enumeration.

   .. grid-item-card:: Global Fields
      :link: GlobalFields/index-global-fields
      :link-type: doc

      Number fields and function fields: rings of integers, ideals, class groups and units.

   .. grid-item-card:: Local Fields
      :link: LocalFields/index-local-fields
      :link-type: doc

      p-adic fields and formal power series: completions, extensions and Newton polygons.

   .. grid-item-card:: Finite Groups
      :link: FiniteGroups/index-finite-groups
      :link-type: doc

      Permutation groups, matrix groups and generic finite groups with structural algorithms.

   .. grid-item-card:: Finitely Presented Groups
      :link: FinitelyPresentedGroups/index-finitely-presented-groups
      :link-type: doc

      Groups given by generators and relations, coset enumeration and rewriting systems.

   .. grid-item-card:: Representation Theory
      :link: RepresentationTheory/index-representation-theory
      :link-type: doc

      Characters, modules over group algebras, and ordinary and modular representations.

   .. grid-item-card:: Lie Theory
      :link: LieTheory/index-lie-theory
      :link-type: doc

      Lie algebras, root systems, Coxeter groups and Chevalley groups.

   .. grid-item-card:: Modules
      :link: Modules/index-modules
      :link-type: doc

      Modules over rings, free and finitely generated modules, and homological constructions.

   .. grid-item-card:: Algebras
      :link: Algebras/index-algebras
      :link-type: doc

      Associative and non-associative algebras, quaternion algebras and crossed products.

   .. grid-item-card:: Commutative Algebra
      :link: CommutativeAlgebra/index-commutative-algebra
      :link-type: doc

      Ideals in polynomial rings, Gröbner bases, primary decomposition and schemes.

   .. grid-item-card:: Algebraic Geometry
      :link: AlgebraicGeometry/index-algebraic-geometry
      :link-type: doc

      Schemes, varieties, divisors and sheaves over general fields.

   .. grid-item-card:: Arithmetic Geometry
      :link: ArithmeticGeometry/index-arithmetic-geometry
      :link-type: doc

      Elliptic and hyperelliptic curves, Jacobians, rational points and descent.

   .. grid-item-card:: Modular Arithmetic Geometry
      :link: ModularArithmeticGeometry/index-modular-arithmetic-geometry
      :link-type: doc

      Modular forms, modular curves and related L-functions.

   .. grid-item-card:: Topology
      :link: Topology/index-topology
      :link-type: doc

      Simplicial complexes, homology groups and topological invariants.

   .. grid-item-card:: Geometry
      :link: Geometry/index-geometry
      :link-type: doc

      Projective and affine geometry, polytopes and incidence structures.

   .. grid-item-card:: Combinatorics
      :link: Combinatorics/index-combinatorics
      :link-type: doc

      Graphs, designs, permutation groups on sets and combinatorial enumeration.

   .. grid-item-card:: Coding Theory
      :link: CodingTheory/index-coding-theory
      :link-type: doc

      Linear codes, decoding algorithms and bounds on minimum distance.

   .. grid-item-card:: Cryptography
      :link: Cryptography/index-cryptography
      :link-type: doc

      Cryptographic primitives and number-theoretic cryptographic algorithms.

   .. grid-item-card:: Optimization
      :link: Optimization/index-optimization
      :link-type: doc

      Integer programming, lattice problems and optimization over algebraic structures.