Sydney: Magma (computer Algebra, Version 2.19, April 24), 2013. — 5488 (cxxxiv+5291+lxiii) p. Interactive menu.Preface. The computer algebra system Magma is designed to provide a software environment for computing with the structures which arise in areas such as algebra, number theory, algebraic geometry and (algebraic) combinatorics. Magma enables users to define and to compute with structures such as groups, rings, fields, modules, algebras, schemes, curves, graphs, designs, codes and many others. Contents.. The magma language. Statements and expressions. Functions, procedures and packages. Input and output. Environment and options. Magma semantics. The magma profiler. Debugging magma code. Sets, sequences, and mappings. Introduction to aggregates. Sets. Sequences. Tuples and cartesian products. Lists. Associative arrays. Coproducts. Records. Mappings. Basic rings.. Introduction to rings. Ring of integers. Integer residue class rings. Rational field. Finite fields. Nearfields. Univariate polynomial rings. Multivariate polynomial rings. Real and complex fields. Matrices and linear algebra. Matrices. Sparse matrices. Vector spaces. Polar spaces. Lattices and quadratic forms. Lattices. Lattices with group action. Quadratic forms. Binary quadratic forms. Global arithmetic fields. Number fields. Quadratic fields. Cyclotomic fields. Orders and algebraic fields. Galois theory of number fields. Class field theory. Algebraically closed fields. Rational function fields. Algebraic function fields. Class field theory for global function fields. Artin representations. Local arithmetic fields.. Valuation rings. Newton polygons. p-Adic rings and their extensions. Galois rings. Power, laurent and puiseux series. Lazy power series rings. General local fields. Algebraic power series rings. Modules.. Introduction to modules. Free modules 1395. modules over dedekind domains. Chain complexes. Finite groups.. Groups. Permutation groups. Matrix groups over general rings. Matrix groups over finite fields. Matrix groups over infinite fields. Matrix groups over q and z. Finite soluble groups. Black-box groups. Almost simple groups. Databases of groups. Automorphism groups. Cohomology and extensions. Finitely-presented groups. Abelian groups. Finitely presented groups. Finitely presented groups: advanced. Polycyclic groups. Braid groups. Groups defined by rewrite systems. Automatic groups. Groups of straight-line programs. Finitely presented semigroups. Monoids given by rewrite systems. Algebras.. Algebras. Structure constant algebras. Associative algebras. Finitely presented algebras. Matrix algebras. Group algebras. Basic algebras. Quaternion algebras. Algebras with involution. Clifford algebras. Representation theory. Modules over an algebra. K[G]-Modules and group representations. Characters of finite groups. Representations of symmetric groups. Mod p galois representations. Lie theory.. Introduction to lie theory. Coxeter systems. Root systems. Root data. Coxeter groups. Reflection groups. Lie algebras. Kac-moody lie algebras. Quantum groups. Groups of lie type. Representations of lie groups and algebras. Commutative algebra.. Grobner bases. Polynomial ring ideal operations. Local polynomial rings. Affine algebras. Modules over multivariate rings. Invariant theory. Differential rings. Algebraic geometry.. Schemes. Coherent sheaves. Algebraic curves. Resolution graphs and splice diagrams. Algebraic surfaces. Hilbert series of polarised varieties. Toric varieties. Arithmetic geometry.. Rational curves and conics. Elliptic curves. Elliptic curves over finite fields. Elliptic curves over q and number fields. Elliptic curves over function fields. Models of genus one curves. Hyperelliptic curves. Hypergeometric motives. L-functions. Modular arithmetic geometry. Modular curves. Small modular curves. Congruence subgroups of psl2(r). Arithmetic fuchsian groups and shimura curves. Modular forms. Modular symbols. Brandt modules. Supersingular divisors on modular curves. Modular abelian varieties. Hilbert modular forms. Modular forms over imaginary quadratic fields. Admissible representations of gl2(qp). Topology. Simplicial homology. Geometry. Finite planes. Incidence geometry. Convex polytopes and polyhedra. Combinatorics.. Enumerative combinatorics. Partitions, words and young tableaux. Symmetric functions. Incidence structures and designs. Hadamard matrices. Graphs. Multigraphs. Networks. Coding theory. Linear codes over finite fields. Algebraic-geometric codes. Low density parity check codes. Linear codes over finite rings. Additive codes. Quantum codes. Cryptography.. Pseudo-random bit sequences. Optimization.. Linear programming.
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