The Connections Project

Abstract Algebra

Various structures studied in real analysis have inherent algebraic structure as well. Thus, the former profits from an understanding of the latter, while the latter can draw many interesting examples from the former.

The very definition of the complex numbers traditionally comes from abstract algebra: you take the ring of real polynomials, mod out by the ideal generated by x^2+1, and since x^2+1 is irreducible, theorems of algebra guarantee the resulting quotient ring is actually a field-- a field with a square root of -1. Complex analysis generously repays the favor by providing an unexpected proof of the Fundamental Theorem of Algebra (which states that every non-constant polynomial with complex coefficients has a root).

When topology and algebra meet, the whole field of algebraic topology is born, and this is easily among the most fascinating areas of higher math. Basically, topologists noticed that there is a certain equivalence between certain families of curves, and when they "modded out" by that equivalence class, a rich and deep algebraic structure was revealed in those curves.

Like real analysis, many of the "algebraic" constructions are essentially just set-theoretic constructions; the biggest example being the operation of "modding out" over an equivalence relation. A decent grasp of set theory is essential to deeply understand, for example, the transition from the standard integers to the integers mod N.

In turn, set theory benefits from algebra because you can treat sets as an algebraic structure, with union and intersection being analogous to addition and multiplication.

One of the biggest sources of examples in abstract algebra is modular arithmetic, the very bread and butter of number theory. Indeed, it's hard to even say whether modular arithmetic falls under the category of algebra or whether it falls more under number theory: really, it's both.

The most famous connection between computability theory and abstract algebra is the celebrated Word Problem: in a group generated by certain generators which are constrained by certain relations, determine when two words are the same. Computability theory is used to show that no general algorithm exists to perform this task.

There is an entire field called algebraic graph theory. For example, graph theory naturally gives rise to questions of symmetry, and these are best explored with the machinery of group theory.

Model theory provides generalizations, as well as alternate proofs, for certain advanced theorems from algebra.

Linear algebra is strictly a subset of abstract algebra. Nonetheless, it's worth mentioning that in a certain sense, vector spaces are "perverse" in an algebraic sense: vectors and scalars can't be added to each other, a lurid state of affairs sure to vex any universal algebraist.

To many students, an encounter with the notion of linear transformations in an elementary linear algebra course is the first sampling of the rich beauty of homomorphisms in general.

All the major classes of structures in abstract algebra are examples of categories: the category of groups, the category of abelian groups, the category of rings, and so on.

At a more advanced level, many algebraic constructions are best defined by use of "universal diagrams", which notion is inherently categorical. See, for example, the abstract definition of the tensor product or even the direct product.