Domain 03 · Mathematics Codex
Algebra
From solving equations to the abstract structures that underlie all of mathematics.
Scope
Elementary algebra, linear algebra, abstract algebra (groups, rings, fields, modules), Galois theory, commutative algebra, homological algebra, Lie algebras and Lie groups, representation theory.
Algebra is the study of structure. When a physicist writes down the symmetry group of a particle, when a cryptographer works with finite fields, when a data scientist multiplies matrices — they are all doing algebra. The abstract structures discovered in the 19th century (groups, rings, fields) turned out to be exactly the right language for quantum mechanics, coding theory, and modern geometry.
Entries in this domain
Build status: This domain is in active build. Live entries are marked below. All other entries are planned and pending — see build status for the full picture.
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Group
Pending
A set with an associative binary operation, an identity element, and inverses — the simplest structure capturing symmetry.
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Ring
Pending
A set with two operations (addition and multiplication) satisfying distributivity — the abstract structure of arithmetic.
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Vector Space
Pending
A set of vectors with addition and scalar multiplication, satisfying eight axioms — the foundation of linear algebra.
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Galois Theory
Pending
The deep connection between field extensions and group theory that explains why polynomial equations of degree 5 or higher have no general formula.
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Linear Map
Pending
A function between vector spaces that preserves addition and scalar multiplication.