Algebra

The study of mathematical structure through groups, rings, fields, and their operations.

• A group is a set with an associative binary operation, identity element, and inverses
• A ring extends groups with a second operation (addition and multiplication)
• A field is a ring where every nonzero element has a multiplicative inverse
• homomorphisms are structure-preserving maps between algebraic objects
• Galois theory connects field extensions to group symmetry, resolving solvability of polynomials
• Boolean algebra underpins logic, digital circuits, and set theory
• Foundation of cryptography, number theory, linear algebra, and category theory
• Related: geometry, topology, combinatorics, game theory