>>12038751Abstract algebra is the formal mathematical study of algebraic structures, which in turn are sets equipped with operations and rules for combining them. This allows mathematicians to study particular structures abstracted away from their specific characteristics so that they can glean a deeper understanding of how they work and how they're related to other algebraic structures. This is important because oftentimes a proof depends on showing that a given mathematical object is really another, seemingly different mathematical object in disguise.
Group theory is a branch of abstract algebra that studies groups; groups in turn are algebraic structures equipped with binary operations such that the result of applying the binary operation to any two elements of the set results in an element in that set. These structures are tremendously important for any study involving symmetry, which include art, physics, chemistry, cryptography, and many other, more specialized branches of mathematics.
Field theory is a branch of abstract algebra that studies fields; fields in turn are algebraic structures that are equipped with two binary operations - addition and multiplication - as well as an additive inverse and a multiplicative inverse for every element in it. These are incredibly important for number theory, geometry, and coding theory.
Galois theory is a theoretical framework that establishes connections between field theory and group theory; it allows the practitioner to reduce complex problems in field theory to simpler problems in group theory.
