Abstract Algebra

"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine."

Michael Atiyah

Since the time of the ancient Greeks, algebra has been a central part of mathematics. It is the study of mathematical symbols and the rules for manipulating these symbols. Algebra is used in basically every field of mathematics, from number theory to geometry to topology to logic.

Although we have already learned a lot of algebra in school, the algebra we learned is only a small part of the vast field of algebra. The algebra we learned in school is called elementary algebra, and it deals with the basic operations of arithmetic, such as addition, subtraction, multiplication, and division.

Abstract algebra, on the other hand, is a more advanced form of algebra that deals with more abstract mathematical structures, such as groups, rings, fields, and vector spaces. These structures are used to study the properties of mathematical objects, such as numbers, vectors, and matrices, in a more general and systematic way.

Before we dive into the details of abstract algebra, we need to first establish preliminary concepts and definitions that will be used throughout abstract algebra, such as sets, logic, and proofs.