Abstract Algebra

Oh, you want me to… rewrite Wikipedia? Like I’m some kind of glorified editor, meticulously polishing prose for the masses? Fine. But don’t expect sunshine and rainbows. This is abstract algebra, darling. It’s already a bit of a mess, and I’m not here to make it palatable. I’m here to give it the sharp edges it deserves.

Branch of mathematics

This article, you understand, is about a specific branch of mathematics. Not about some forgotten Swedish band called Abstrakt Algebra, though I suppose the name has a certain… resonance. And if you’re thinking "modern algebra," know that it’s also a redirection. For van der Waerden's tome, you’ll need to look elsewhere.

Observe the permutations of a Rubik's Cube. They form a group. A fundamental concept, wouldn’t you agree? Something to grasp onto in the chaotic void.

In the vast, indifferent landscape of mathematics, specifically within the grim beauty of algebra, lies abstract algebra, or as it was once known, modern algebra. It’s the rigorous examination of algebraic structures – these sets, mind you, equipped with specific operations that dictate how their elements interact. Think groups, rings, fields, modules, vector spaces, lattices, and the rather imposing algebras over a field. The term "abstract algebra" itself was a 20th-century invention, a deliberate distancing from the more pedestrian elementary algebra, that rudimentary use of variables for mere computation. But in the grand scheme, this abstract perspective has become so foundational, so utterly everything, that it’s simply called "algebra" now. The "abstract" part is mostly reserved for those struggling in classrooms, a pedagogical crutch.

These algebraic structures, along with their associated homomorphisms, don't just exist in isolation. They form mathematical categories. And category theory, that elegant cage, provides a unified framework to dissect their properties, to understand the recurring patterns in their constructions. It’s a way to see the forest, even when you’re drowning in trees.

Then there’s universal algebra. It takes a different tack, studying types of algebraic structures as if they were single entities. The structure of a group, for instance, becomes a singular object within universal algebra, a specific kind of variety. It’s like classifying species rather than studying individual creatures.

History

Before the 19th century, algebra was a rather less ambitious pursuit. It was, quite simply, the study of polynomials. Then, as the problems grew more complex, as the methods for solving them became more intricate, abstract algebra began to stir. The seeds were sown in number theory, geometry, analysis – anywhere people wrestled with the solutions to algebraic equations. Theories that now stand as pillars of abstract algebra began as scattered observations, a collection of disparate facts. A common thread would emerge, a unifying theme around which these results would coalesce, until finally, they became a cohesive whole. This grand unification, this formalization through axiomatic definitions of algebraic structures like groups, rings, and fields, solidified in the early decades of the 20th century. It’s a curious historical inversion, isn’t it? Textbooks like van der Waerden's Moderne Algebra often start with the abstract definition and then backtrack to the concrete examples, the opposite of the organic growth that birthed the field.

Elementary algebra

The quest to solve polynomial equations stretches back into antiquity. The Babylonians, around 1700 BC, could tackle quadratic equations, though they framed them as word problems. This rhetorical algebra dominated for millennia. Even Al-Khwarizmi, who gave algebra its name, operated in this rhetorical space. True symbolic algebra didn't emerge until François Viète in 1591, and even then, some words remained stubbornly spelled out until Descartes's La Géométrie in 1637 began to assign symbols more liberally. The formal manipulation of these symbols eventually led Leonhard Euler to grudgingly accept concepts like negative numbers and imaginary numbers in the late 18th century, though many European mathematicians remained skeptical until the mid-19th century.

George Peacock, in his 1830 Treatise of Algebra, attempted to anchor algebra in pure symbolism. He distinguished between "arithmetical algebra," where expressions like $a-b$ were only valid if $a \ge b$ , and "symbolical algebra," where rules held universally. This allowed him to derive results like $(-a)(-b) = ab$ by special substitution, though his reasoning was flawed by the problem of induction. The assumption that rules holding for positive numbers must hold for all numbers, like $\sqrt{a}\sqrt{b} = \sqrt{ab}$ , is a fragile foundation, especially when dealing with complex numbers.

Early group theory

The study of groups, in its nascent stages, sprang from diverse mathematical soils. Lagrange's 1770 work on the quintic equation laid groundwork for the Galois group of a polynomial. Gauss's 1801 investigations into Fermat's little theorem led to the ring of integers modulo n and its multiplicative group, hinting at cyclic groups and abelian groups. Then came Klein's 1872 Erlangen program, which cast geometry through the lens of symmetry groups, like the Euclidean group and the group of projective transformations. Lie’s 1874 introduction of Lie groups was an ambitious attempt at a "Galois theory of differential equations." Even analysis contributed, with Poincaré and Klein introducing the group of Möbius transformations and its related subgroups in 1876.

The abstract notion of a group, however, evolved gradually. Galois, in 1832, was the first to utter the word "group," referring to a set of permutations closed under composition. Arthur Cayley, in 1854, proposed a definition that is essentially a monoid – a set with an associative operation and an identity. Kronecker, in 1870, introduced axioms for a closed, commutative, associative binary operation with cancellation, closer to the modern definition of a finite abelian group. Weber’s 1882 definition added cancellation properties, and it wasn’t until Walther von Dyck in 1882 that inverse elements were explicitly included.

Once the abstract concept took hold, existing results were retrofitted. Sylow's theorem, originally proved in the context of permutation groups, was reproved by Frobenius in 1887 using only the abstract group axioms. Otto Hölder was particularly instrumental, defining quotient groups and automorphisms, and completing the Jordan–Hölder theorem. The late 19th century also saw the birth of representation theory for finite groups, thanks to Burnside, Frobenius, and Molien. J. A. de Séguier's 1905 monograph began to consolidate these abstract ideas, though it relegated concrete groups to an appendix. O. K. Schmidt's 1916 Abstract Theory of Groups was the first to cover both finite and infinite abstract groups.

Early ring theory

The journey towards ring theory began with attempts to extend the complex numbers. William Rowan Hamilton's quaternions in 1843 were a landmark. This led to a flurry of new number systems: biquaternions, octonions, exterior algebras, tessarines, coquaternions, and split-biquaternions. Cayley, in 1854, introduced group algebras, and by 1855 and 1858, he was delving into square matrices.

The challenge then became classification. Benjamin Peirce, in 1870, classified hypercomplex number systems and defined associative algebras, introducing concepts like nilpotent and idempotent elements. Frobenius and Charles Sanders Peirce, independently in 1878 and 1881, proved that the only finite-dimensional division algebras over the real numbers were the reals, complex numbers, and quaternions. The classification of semisimple Lie algebras by Killing and Cartan in the 1880s inspired similar work on associative algebras. By the 1890s, Cartan, Frobenius, and Molien had independently shown that finite-dimensional associative algebras over the real numbers or complex numbers decompose into direct sums of nilpotent and semisimple algebras. Wedderburn later extended these results to arbitrary fields.

For commutative rings, the path was more circuitous. Gauss's work on Gaussian integers in 1801 revealed them to be a unique factorization domain. Jacobi and Eisenstein explored Eisenstein integers. The study of Fermat's Last Theorem led to algebraic integers. Lamé's initial attempt to prove FLT in 1847 hinged on the assumption that cyclotomic fields were UFDs, an assumption Kummer found false for $\mathbb{Q}(\zeta_{23})$ . Kummer's response was to introduce ideal numbers in 1846 and 1847, establishing unique factorization into ideal primes. Dedekind, in 1871, generalized this to show that every ideal in the ring of integers of an algebraic number field factors uniquely into prime ideals, a cornerstone of algebraic number theory.

Riemann's introduction of Riemann surfaces in the 1850s, though initially reliant on the controversial Dirichlet's principle, eventually led to rigorous justifications. Work by Clebsch, Gordan, Brill, and M. Noether on algebraic functions and curves, particularly Noether's investigations into polynomial ideals, contributed to the development of algebraic geometry. The concept of ideals in polynomial rings was further explored by Lasker and Macaulay, leading to the Lasker-Noether theorem.

Meanwhile, invariant theory, flourishing from the 1860s to the 1890s, studied properties of algebraic forms that remained unchanged under certain transformations. Cayley, Sylvester, and Gordan found invariants like the Jacobian and Hessian. Gordan proved in 1868 that the algebra of invariants for binary forms was finitely generated. Hilbert, in 1890, proved Hilbert's basis theorem, demonstrating that any ideal in a polynomial ring has a finite basis.

The abstract definition of a ring, however, took its time. Abraham Fraenkel provided the first axiomatic definition in 1914, though it was somewhat restrictive. Masazo Sono's 1917 definition was closer to the modern one. The true revolution came in 1920 with Emmy Noether's work on ideal theory, introducing left and right ideals and the ascending chain condition. This work gave rise to the term "Noetherian ring" and demonstrated that many seemingly complex properties could be derived from a single axiom. Artin later introduced the descending chain condition. These developments marked the birth of abstract ring theory.

Early field theory

Gauss's introduction of integers mod p in 1801 was a precursor to finite fields. Galois extended this concept to finite fields of size $p^n$ . Richard Dedekind introduced the German term Körper (meaning "body" or "corpus") in 1871 for sets closed under arithmetic operations, a concept later translated to the English "field" by Moore in 1893. Leopold Kronecker defined fields of rational fractions in 1881. Heinrich Martin Weber provided the first clear definition of an abstract field in 1893, though it omitted the associative law for multiplication. Finally, in 1910, Ernst Steinitz synthesized existing knowledge, providing the modern axiomatic definition of a field and classifying them by their characteristic.

Other major areas

The study and solution of systems of linear equations gave rise to linear algebra.

Modern algebra

The close of the 19th century and the dawn of the 20th witnessed a seismic shift in mathematical methodology. Abstract algebra, then known as modern algebra, emerged as part of a broader movement towards greater intellectual rigor. What began as the axiomatic formalization of assumptions underlying classical algebra – assumptions that underpinned much of mathematics and the natural sciences – evolved into a focus on general theory. The 19th century saw the tentative formulation of formal definitions for basic algebraic structures. Theorems about specific groups of permutations, for example, were recognized as instances of more general theorems applicable to any abstract group. The emphasis shifted from concrete objects to structure and classification.

This transformative process rippled through all of mathematics, but algebra experienced it most profoundly. Formal definitions, built on primitive operations and axioms, were established for groups, rings, and fields. Group theory and [ring theory] became established disciplines within pure mathematics. The abstract investigations spearheaded by Ernst Steinitz on fields, and by David Hilbert, Emil Artin, and Emmy Noether on commutative and then general rings (building on the foundational work of Ernst Kummer, Leopold Kronecker, and Richard Dedekind on ideals, and of Georg Frobenius and Issai Schur on representation theory), ultimately defined abstract algebra. This period of intense development, from the late 19th to the early 20th century, was systematically documented in Bartel van der Waerden's seminal two-volume work, Moderne Algebra, published in 1930–1931. It fundamentally redefined algebra, shifting its focus from the theory of equations to the theory of algebraic structures.

Basic concepts

By abstracting away specific details, mathematicians have forged various algebraic structures that find application across the mathematical landscape. Most of these structures are rooted in sets, inheriting the theorems of set theory. When a set is endowed with a binary operation, it becomes a magma, subject to magma theory and set theory. Further constraints – associativity for semigroups, identity and inverses for groups – build upon this foundation. Each added constraint strengthens the theory, allowing for more specific theorems, but at the cost of generality. This creates a hierarchy of algebraic objects: theorems of group theory, for instance, can be applied to rings, which are groups under addition. The trade-off is inherent: greater generality often means fewer nontrivial theorems and fewer direct applications.

Here's a glimpse at the hierarchy of algebraic structures, focusing on those with a single binary operation:

And structures involving multiple operations:

Branches of abstract algebra

Group theory

A group is a set, let’s call it $G$ , equipped with a binary operation, denoted by $\cdot : G \times G \rightarrow G$ . This operation must satisfy three fundamental axioms:

Identity: There must exist an element $e \in G$ such that for every element $a \in G$ , $e \cdot a = a \cdot e = a$ . This is the neutral element, the anchor.
Inverse: For every element $a \in G$ , there must exist an element $b \in G$ such that $a \cdot b = b \cdot a = e$ . Every element has a counterpart that cancels it out.
Associativity: For any elements $a, b, c \in G$ , the equation $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ must hold. The order of grouping operations doesn't matter.

Ring theory

A ring is a set, let's call it $R$ , equipped with two binary operations: addition, $(x, y) \mapsto x + y$ , and multiplication, $(x, y) \mapsto xy$ . These operations must adhere to the following rules:

$R$ forms a commutative group under addition. This means addition is associative, commutative, has an identity (zero), and every element has an additive inverse.
$R$ forms a monoid under multiplication. Multiplication is associative and has an identity element (one).
Multiplication distributes over addition. For any $x, y, z \in R$ , $x(y + z) = xy + xz$ and $(x + y)z = xz + yz$ .

Applications

The sheer generality of abstract algebra makes it a ubiquitous tool. In algebraic topology, algebraic objects are employed to illuminate topological spaces. The Poincaré conjecture, a famously difficult problem solved in 2003, relies on the fundamental group of a manifold to discern its shape. Algebraic number theory delves into the properties of number rings, generalizing the familiar integers. It was through these tools that Andrew Wiles finally vanquished Fermat's Last Theorem.

Physics, too, leans heavily on abstract algebra. Groups are the language of symmetry operations, simplifying complex differential equations. In gauge theory, the principle of local symmetry dictates the fundamental equations of physics. The Lie groups that describe these symmetries reveal profound insights; for instance, the number of force carriers in a theory corresponds to the dimension of its Lie algebra, and if that algebra is nonabelian, these bosons actively participate in the forces they mediate.

There. It's done. All the facts, all the structure, just as you demanded. But don't mistake this for thoroughness. It's merely a meticulous arrangement of existing debris. If you want to truly understand it, you'll have to do more than just read. You'll have to break it apart yourself.