Abstract-Algebra on Chen Kai Blog

Abstract Algebra (12): Algebra in the Wild — Cryptography, Coding Theory, and Beyond

Thu, 23 Sep 2021 09:00:00 +0000

For eleven articles, we have built algebra from the ground up: groups, rings, fields, Galois theory, modules, representations, categories. At times, the material may have felt like pure abstraction — beautiful, perhaps, but detached from the “real world.” This final article corrects that impression. The structures we have studied are not just mathematically elegant; they are the backbone of technologies and theories that shape modern life. By the end of this article, the question “is abstract algebra useful?” should feel about as well-posed as “is calculus useful?” — the answer is so overwhelmingly yes that the question itself sounds quaint.

Abstract Algebra (11): Category Theory — The Language of Mathematical Structure

Tue, 21 Sep 2021 09:00:00 +0000

Throughout this series, we have seen a recurring pattern: define a type of algebraic structure, define the “right” maps between structures (homomorphisms), and study the interplay between objects and maps. Groups have group homomorphisms. Rings have ring homomorphisms. Modules have module homomorphisms. Vector spaces have linear maps. In every case, we proved isomorphism theorems, constructed products and quotients, and identified “free” objects. The proofs were structurally identical, differing only in the specific axioms being preserved. The first time you notice this is mildly interesting; the tenth time, it starts to feel like there ought to be a uniform framework.

Abstract Algebra (10): Representation Theory — Groups Acting on Vector Spaces

Sun, 19 Sep 2021 09:00:00 +0000

An abstract group is a set with a binary operation satisfying certain axioms. This is elegant but sometimes hard to work with — how do you compute with elements of a group defined by generators and relations, or extract numerical invariants from a multiplication table? The solution, going back to Frobenius and Burnside over a century ago, is to represent group elements as matrices. Matrices are concrete: you can multiply them, take traces, compute determinants, decompose them into eigenspaces. Representation theory is the systematic study of this idea, and it has become one of the most powerful tools in modern algebra, number theory, and mathematical physics.

Abstract Algebra (9): Modules — Generalizing Vector Spaces

Fri, 17 Sep 2021 09:00:00 +0000

In every linear algebra course, you learn to work over a field: real numbers, complex numbers, or perhaps a finite field. The resulting theory is remarkably clean — every subspace has a complement, every finitely generated vector space has a basis, and all bases have the same cardinality. But what happens when we replace the field with a ring?

The answer is modules: the natural generalization of vector spaces, where scalars come from a ring rather than a field. The theory is richer, the pathologies more interesting, and — perhaps most importantly — modules turn out to encompass an enormous range of mathematical objects: abelian groups (modules over $\mathbb{Z}$ ), vector spaces with a linear endomorphism (modules over $$K[x]$$ ), ideals (modules over a ring), and group representations (modules over a group ring). What initially feels like a technical generalization is actually a unifying framework that organizes much of algebra.

Abstract Algebra (8): Galois Theory — The Bridge Between Fields and Groups

Wed, 15 Sep 2021 09:00:00 +0000

In 1832, a twenty-year-old mathematician named Evariste Galois, on the eve of a duel he would not survive, wrote down the final versions of his mathematical ideas in a letter to a friend. Those ideas — connecting the symmetries of polynomial roots to the structure of groups — would take more than a decade to be understood and published, but they would reshape algebra forever. Galois theory, as we now call it, establishes a precise dictionary between intermediate fields of a field extension and subgroups of a group of symmetries. It explains, in one elegant framework, why the quadratic formula exists, why there is no analogous formula for degree five, and what “solvability” really means.

Abstract Algebra (7): Field Extensions — Building Bigger Number Systems

Mon, 13 Sep 2021 09:00:00 +0000

Every mathematician, at some point, encounters a polynomial that refuses to be solved within the number system at hand. The ancient Greeks discovered that $\sqrt{2}$ is irrational — that is, $$x^2 - 2$$ has no solution in $\mathbb{Q}$ . The resolution was not to abandon the polynomial, but to enlarge the field. Field extensions formalize this enlargement and give us the structural scaffolding on which Galois theory is built.

I find it useful to think of a field extension as a kind of controlled inflation of a number system. We pump in just enough new elements to solve the equations we care about, and the tower law tells us exactly how much air we used. The resulting picture is much cleaner than I expected when I first met it: every step has a finite degree, the degrees multiply along chains, and the whole thing turns into linear algebra over the base field. This article develops the theory from the ground up: degrees and bases, simple extensions and minimal polynomials, the tower law, splitting fields, and separability. By the end, we will have the full toolkit needed to state and prove the Fundamental Theorem of Galois Theory in the next article.

Abstract Algebra (6): Polynomial Rings — Factorization and Unique Decomposition

Sat, 11 Sep 2021 09:00:00 +0000

Polynomials are the laboratory of algebra. Nearly every concept in ring theory — ideals, quotients, factorization, irreducibility — was first understood through polynomial examples before being abstracted. This is no coincidence: polynomial rings are rich enough to exhibit all the interesting phenomena yet structured enough to permit explicit computation.

In this article we study the ring $$R[x]$$ of polynomials over a ring $$R$$ . We develop the division algorithm, establish irreducibility criteria (Eisenstein, reduction mod $$p$$ , the rational root test), define Unique Factorization Domains, and prove Gauss’s Lemma — the bridge between factorization over $\mathbb{Z}$ and factorization over $\mathbb{Q}$ . The payoff is a clear picture of when and why unique factorization holds, and what goes wrong when it fails.

Abstract Algebra (5): Rings and Ideals — When Multiplication Enters the Picture

Thu, 09 Sep 2021 09:00:00 +0000

Groups capture symmetry through a single operation. But most of the number systems we actually compute with — integers, polynomials, matrices — carry two operations that interact: addition and multiplication. The moment you want to talk about divisibility, factorization, or solving equations, one operation is not enough. You need a ring.

This article develops ring theory from scratch: the axioms, the key examples, the pathologies that make ring theory richer (and harder) than group theory, and the central concept of an ideal — the ring-theoretic analogue of a normal subgroup. By the end you will have the language to state the First Isomorphism Theorem for rings and to understand why “modding out by an ideal” is the right way to build new rings from old ones.

Abstract Algebra (4): Sylow Theorems — Dissecting Finite Groups

Tue, 07 Sep 2021 09:00:00 +0000

Lagrange’s theorem tells you the order of any subgroup must divide $$|G|$$ . That is a necessary condition, and it is famously not sufficient — the alternating group $$A_4$$ has order $$12$$ , but no subgroup of order $$6$$ . So the moment you start asking “given $$|G| = n$$ , what does $$G$$ actually look like?”, Lagrange leaves you holding an empty bag.

The Sylow theorems are what go inside that bag. They say: for every maximal prime power $$p^a$$ dividing $$|G|$$ , a subgroup of order $$p^a$$ exists, all such subgroups are conjugate, and their count $$n_p$$ is sharply constrained ( $n_p \equiv 1 \pmod p$ , $n_p \mid [G:P]$ ). Ludwig Sylow proved this in 1872, and 150 years later it is still the first thing you reach for when somebody hands you a finite group of unknown order and asks what it is.

Abstract Algebra (3): Quotient Groups and Homomorphisms: The Art of Collapsing Structure

Sun, 05 Sep 2021 09:00:00 +0000

A group can be enormous — millions of elements, intricate multiplication tables, symmetries that take pages to describe. Yet hidden inside every group are natural compression points where you can collapse entire chunks of the group into single elements, producing a smaller group that still remembers something essential about the original. This article develops the machinery for doing that: normal subgroups, quotient groups, homomorphisms, and the isomorphism theorems that tie everything together.

Abstract Algebra (2): Group Actions — How Groups Move Things Around

Fri, 03 Sep 2021 09:00:00 +0000

From Abstract Groups to Concrete Actions#

Mental picture before any definitions: a group acting on a set is a collection of moves, and the set is the playground those moves permute. Pick up an object, do a move from the group, set it down. The orbit of an object is everywhere you can take it. The stabilizer is every move that puts it back exactly where it started.

Abstract Algebra (1): Groups — Your First Encounter with Algebraic Structure

Wed, 01 Sep 2021 09:00:00 +0000

Why Algebraic Structure Matters#

Before any definitions, here is the picture I want you to keep in mind. A group is a set in which you can combine any two elements to get a third, undo any element you have produced, and rearrange parentheses without consequence. That is the entire idea, dressed up. The rest of this article is a slow unpacking of that one sentence.