The Magma Handbook#
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.
About Magma and the handbook
Alphabetical index of all functions
Bibliography
Syntax, types, control flow, procedures, functions and the Magma programming model.
Fundamental aggregate types: sets, multisets, sequences, tuples and mappings.
Integers, rationals, finite fields, residue class rings and polynomial rings.
Matrix construction, arithmetic, decompositions and solving linear systems.
Integral lattices, quadratic forms, reduction algorithms and enumeration.
Number fields and function fields: rings of integers, ideals, class groups and units.
p-adic fields and formal power series: completions, extensions and Newton polygons.
Permutation groups, matrix groups and generic finite groups with structural algorithms.
Groups given by generators and relations, coset enumeration and rewriting systems.
Characters, modules over group algebras, and ordinary and modular representations.
Lie algebras, root systems, Coxeter groups and Chevalley groups.
Modules over rings, free and finitely generated modules, and homological constructions.
Associative and non-associative algebras, quaternion algebras and crossed products.
Ideals in polynomial rings, Gröbner bases, primary decomposition and schemes.
Schemes, varieties, divisors and sheaves over general fields.
Elliptic and hyperelliptic curves, Jacobians, rational points and descent.
Modular forms, modular curves and related L-functions.
Simplicial complexes, homology groups and topological invariants.
Projective and affine geometry, polytopes and incidence structures.
Graphs, designs, permutation groups on sets and combinatorial enumeration.
Linear codes, decoding algorithms and bounds on minimum distance.
Cryptographic primitives and number-theoretic cryptographic algorithms.
Integer programming, lattice problems and optimization over algebraic structures.