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Abstract algebra learning benefits

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This page lists benefits of learning the subject abstract algebra. In other words, it tries to answer the question Why should I learn abstract algebra? |See all pages on the benefits of learning specific subjects

This page describes some benefits of learning abstract algebra. The goal of the page is to help you better understand whether learning abstract algebra is worth the effort; it is not to sell the idea of learning abstract algebra to readers.

Basic abstract algebra, as covered at the undergraduate level, includes groups, rings, and fields. There are many other more advanced topics within abstract algebra, but this page concentrates on benefits of acquiring the basic foundation.

Key takeaways

  • Abstract algebra is largely useful only for people who intend to major in mathematics and do further work in a mathematics-intensive area (such as physics, some parts of computer science, and some parts of chemistry).
  • Abstract algebra provides a deeper conceptual foundation for linear algebra, which is quite useful in a broader range of subjects.
  • The modes of thinking introduced in abstract algebra, particularly the ideas of isomorphism and homomorphism, can be useful for understanding logic, philosophy, and various cognitive science-based subjects. However, the specifics of algebraic structures are not relevant.

Utility of abstract algebra in mathematics-intensive areas

Within mathematics itself

Groups, rings, and fields appear in almost all parts of mathematics, although the level of depth to which one needs to understand the material varies from subject to subject. For instance:

  • Shallow level: In analysis, the fields of real numbers and complex numbers appear quite a bit. Also, particular algebras over or appear quite a bit in functional analysis, and the concept of "algebra over a field" is a slight variant of the idea of a ring. However, for the most part, only a very basic familiarity with terminology from algebra, rather than a deep knowledge of algebraic constructs, is needed. Similarly, many branches of mathematics use concepts of isomorphism and homomorphism that originate from algebra.
  • Deep level: Research in algebraic geometry, algebraic number theory, representation theory, and to some extent geometric group theory and the related topological questions relies on a deep knowledge of algebraic constructs.

Within physics and parts of chemistry

Groups, in the form of symmetry and transformation groups, play an important role in encapsulating key ideas, both in quantum mechanics and general relativity. Algebraic constructions related to vector spaces and algebras, such as direct sums, structure constants, and tensor products, turn up in specialized contexts in physics.

Some parts of chemistry, such as those dealing with molecular symmetry, require knowledge of some parts of group theory.

Within computer science

Finite fields play an important role within computer science, particularly in the context of cryptography.

Importance in linear algebra

Linear algebra is an extremely important subject that comes up in modeling many situations across the natural and social sciences. Abstract algebra can help with understanding what's going on "under the hood" with linear algebra:

  • Group theory provides the basic framework for understanding how to manipulate expressions involving noncommutative matrix multiplication. Many geometric operations and linear algebra algorithms can be better understood from the group-theoretic perspective.
  • Ring theory provides the basic framework for understanding how to manipulate matrix addition and multiplication together.
  • The theory of modules over rings generalizes that of vector spaces over fields. Contrasting vector spaces with the more general case of modules also provides sufficient contrast to explain just what it is about fields that makes vector spaces so well-behaved.

Utility in terms of general thinking modes

  • The concept of isomorphism -- which essentially means identical from the perspective of the structural aspects we are interested in -- is useful in many parts of philosophy and some parts of cognitive science and economics. Similarly for the concepts of equivalence relation, partial order, and morphism. These concepts can be learned without studying abstract algebra, but abstract algebra provides the most natural setting for learning and internalizing the ideas.