Abstract algebra

In mathematics, abstract algebra or modern algebra is the study of algebraic structures. These are sets with special operations that work on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. These help mathematicians see patterns and connections in many parts of math.

The name abstract algebra started in the early 1900s. It was used to separate this area from older algebra, called elementary algebra. Elementary algebra uses variables to stand for numbers in math problems. Today, abstract algebra is very important in advanced math, and people often just say "algebra."

Algebraic structures, along with special maps called homomorphisms, are part of mathematical categories. Category theory helps study properties that are the same across different structures. A related area, universal algebra, looks at types of algebraic structures as single objects. For example, the structure of groups is studied as one object, called the variety of groups.

History

Before the nineteenth century, algebra was mostly about solving problems with numbers. Abstract algebra started to grow in the 1800s when mathematicians met new and harder problems. Thoughts from number theory, geometry, and solving equations helped create this new way of thinking.

Later, mathematicians saw that many problems had the same patterns. They studied these patterns in a general way. This led to the formal ideas of groups, rings, and fields in the early 1900s. This change moved algebra from just solving equations to looking at the structures themselves.

Main article: History of algebra

Basic concepts

Main article: Algebraic structure

Mathematicians study different ways to group and combine things. These things are called sets, which are just collections of items. The ways we combine them are called operations. For example, adding or multiplying numbers are operations.

When we add rules to these operations, we get different kinds of math structures. Some simple ones are magmas, semigroups, and groups.

More complex structures, like rings and fields, have extra operations and rules. Each kind of structure has its own special properties and uses in math. This helps mathematicians find patterns and solve problems in many areas.

Examples of algebraic structures include:

• Magma
• Quasigroup
• Monoid
• Semigroup
• Group
• Ring
• Field
• Module
• Vector space
• Algebra over a field
• Associative algebra
• Lie algebra
• Lattice
• Boolean algebra

Branches of abstract algebra

Group theory

Main article: Group theory § Definition

Group theory is a part of abstract algebra that studies groups. A group is a set of things where you can combine any two things and get another thing from the same set. For a group to work, it must follow three rules: there must be a special thing that doesn't change anything when combined with other things, every thing must have a matching thing that brings it back to the special thing when combined, and the way you combine things must work the same way no matter the order.

Ring theory

Main article: Ring theory

Ring theory is another part of abstract algebra that studies rings. A ring is a set of things where you can add and multiply any two things and get another thing from the same set. For a ring to work, it must follow three rules: when you add things, the order doesn't matter, there must be a special thing that doesn't change anything when added to other things, and multiplication must work with addition in a specific way.

Applications

Abstract algebra is a useful tool in many areas of mathematics and science. It works with general ideas that can be used in many places.

For example, in a branch called algebraic topology, mathematicians use special algebra tools to study shapes and their properties. One famous result in this area is the Poincaré conjecture. It helps figure out if a shape is a sphere by looking at its fundamental group.

In physics, groups from abstract algebra help describe patterns in nature. These patterns can make complex equations easier and help scientists understand important forces. For instance, in gauge theory, the idea of local symmetry helps find the equations that describe physical systems. The study of special groups called Lie groups can show important details about these physical systems.