Abstract algebra

Abstract algebra studies sets equipped with operations that satisfy specified axioms. Its central structures include groups, rings, and fields.

Group: A set with an associative operation, an identity element, and an inverse for every element. Groups isolate the behavior of a single operation and form a central structure in the subject.
Binary operation: A rule assigning each pair of elements in a set to an element of that set. An operation supplies the basic behavior that structure axioms constrain.
Group representation: A way of expressing group elements as linear transformations of a vector space. Representations convert abstract symmetry into matrices that can be calculated with.
Elementary algebra: The use of symbols and equations to represent quantities and solve for unknowns. It focuses on manipulating expressions, while abstract algebra studies the laws behind operations.
Évariste Galois: A French mathematician whose work connected polynomial equations with permutation groups. His ideas established a deep link between equation solvability and group structure.
Ring: A set with addition and multiplication satisfying specified distributive and additive-group laws. Rings extend group ideas to two interacting operations, including familiar number systems.
Associativity: The law that changing the grouping of repeated operations does not change the result. Associativity is a defining law for groups and a common axiom in algebraic systems.
Galois theory: The study of relationships between field extensions and groups of field automorphisms. It uses groups to explain when polynomial equations can be solved by radicals.
Linear algebra: The study of vector spaces, linear maps, and systems of linear equations. It specializes in vector spaces and linear transformations rather than algebraic structures broadly.
Arthur Cayley: A British mathematician who helped develop group theory and introduced the abstract group concept. His work helped make groups objects studied independently of particular transformations.

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