Short answers, pulled from the story.
Abstract algebra, also called modern algebra, is the study of algebraic structures, which are sets paired with specific operations acting on their elements. These structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field.
The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, especially elementary algebra. The abstract perspective is now so central to advanced mathematics that it is simply called algebra, with the term abstract algebra used mainly in teaching.
Galois in 1832 was the first to use the term group, signifying a collection of permutations closed under composition. Arthur Cayley's 1854 paper On the theory of groups defined a group as a set with an associative composition and an identity, a structure now called a monoid.
Noncommutative ring theory began with William Rowan Hamilton's quaternions in 1843, which extended the complex numbers to hypercomplex numbers. The first axiomatic definition of a ring came from Abraham Fraenkel in 1914, and Masazo Sono's 1917 definition was the first equivalent to the modern one.
Richard Dedekind introduced the German word Körper in 1871 for a set of real or complex numbers closed under the four arithmetic operations, and Moore introduced the English term field in 1893. Heinrich Martin Weber gave the first clear definition of an abstract field in 1893, and Steinitz synthesized abstract field theory in 1910.
In physics, groups represent symmetry operations and can simplify differential equations, while Lie groups and Lie algebras describe symmetries whose dimension equals the number of force carriers in a theory. Andrew Wiles used algebraic number theory to prove Fermat's Last Theorem, and the Poincaré conjecture, proved in 2003, uses the fundamental group of a manifold to determine whether it is a sphere.