Abstract Algebra
What is abstract algebra?
Abstract algebra is a branch of mathematics focused on the study of algebraic structures.
An algebraic structure consists of a set of elements S and a set of operations (e.g., +, *) defined on these elements.
It is called a "structure" because it is characterized by a set and at least one composition law (binary operation).
For example, this algebraic structure consists of the set of natural numbers N={1,2,3,...} and the multiplication operation * $$ (N,*) $$
The simplest algebraic structure is called a magma.
From the concept of a magma, various types of algebraic structures emerge based on the properties of the composition law.
The primary algebraic structures studied in abstract algebra include:
- rings
- groups
- fields
- division ring
- vector spaces
- lattices
Abstract algebra also underpins Boolean algebra and linear algebra.
Why is it called abstract algebra? It is termed "abstract" because it defines the properties of algebraic structures independently of any specific numerical representation. The term "abstract algebra" was coined in the early 20th century to distinguish this axiomatic branch of mathematics from elementary algebra, which deals with the rules for performing calculations within the set of real and complex numbers.
What is the purpose of abstract algebra?
It has numerous practical applications in both scientific and technological fields.
For example, linear algebra, a discipline within abstract algebra, is used in various fields such as engineering and physics.
Moreover, abstract algebra forms the foundation of modern mathematics and contributes significantly to the development of new theories.
The history of abstract algebra. Abstract algebra emerged in the 19th century to address complex problems in various branches of mathematics, such as Gauss's work on fields and Galois's studies on polynomial groups. In the early 20th century, these theories were consolidated into a single discipline named "abstract algebra" to distinguish it from elementary algebra. Throughout the 20th century, axiomatic definitions of various algebraic structures were developed, and these are now studied in university courses in mathematics, physics, and engineering.
And so on.
