Abstract algebra

• 1. Introduction

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  Abstract algebra is the study of algebraic structures, sets paired with specific operations that act on their elements. Groups, rings, fields, modules, vector spaces, lattices, and algebras over a field all belong to this family. The term itself was coined in the early 20th century to mark a break from older parts of algebra, especially the elementary kind where variables stand in for numbers. The abstract view has grown so central to advanced mathematics that practitioners now simply call it algebra. The qualifier abstract survives mostly in teaching. So how did a subject built from disparate facts across number theory, geometry, and analysis come to rest on a handful of axioms? Why did the people who built it spend nearly a century arguing about what a group even was? And why does a textbook open with a definition when history ran almost exactly the other way?

• 2. The Long Road From Polynomials

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  Circa 1700 BC, the Babylonians solved quadratic equations posed as word problems. This was rhetorical algebra, where everything is spelled out in language rather than symbols, and it dominated the field up to the 16th century. Al-Khwarizmi gave algebra its name in 830 AD, yet his work too was entirely rhetorical. Fully symbolic algebra did not arrive until François Viète's New Algebra of 1591, and even that still carried words later turned into symbols in Descartes's La Géométrie of 1637. The push toward symbols forced mathematicians to confront strange new objects. Leonhard Euler, in the late 18th century, accepted what were then dismissed as nonsense roots, including negative numbers and imaginary numbers. Most European mathematicians resisted such ideas until the middle of the 19th century. George Peacock's Treatise of Algebra of 1830 was the first attempt to set algebra on a strictly symbolic basis, separating a new symbolical algebra from the old arithmetical kind. Peacock leaned on what he called the principle of the permanence of equivalent forms, but his reasoning stumbled on the problem of induction. A rule could hold for the nonnegative real numbers and fail for general complex numbers, a gap that hinted at how much rigor the subject still lacked.

• 3. What Is A Group

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  Galois in 1832 was the first to use the word group, meaning a collection of permutations closed under composition. The idea behind it had been gathering for decades from several directions. Lagrange's 1770 study of the quintic equation led toward the Galois group of a polynomial. Gauss's 1801 study of Fermat's little theorem produced the ring of integers modulo n, the multiplicative group of integers modulo n, and the broader notions of cyclic and abelian groups. Klein's Erlangen program of 1872 tied geometry to symmetry groups such as the Euclidean group, and in 1874 Lie introduced the theory of Lie groups, aiming for a Galois theory of differential equations. The abstract definition emerged in fits and starts. Arthur Cayley's 1854 paper On the theory of groups defined a group as a set with an associative composition and an identity, which today is called a monoid. Kronecker in 1870 described a closed, commutative, associative operation with the left cancellation property. Weber's 1882 definition demanded associativity with both left and right cancellation. Walther von Dyck, also in 1882, was the first to insist on inverse elements as part of the definition. Once the abstract concept settled, old results were rebuilt on it. Otto Hölder proved especially prolific, defining quotient groups in 1889, group automorphisms in 1893, and simple groups, while also completing the Jordan-Hölder theorem. J. A. de Séguier's 1905 monograph presented many of these findings in abstract form, banishing concrete groups to an appendix, though it covered only finite groups. The first monograph to treat both finite and infinite abstract groups was O. K. Schmidt's Abstract Theory of Groups of 1916.

• 4. From Quaternions To Noetherian Rings

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  William Rowan Hamilton's quaternions of 1843 opened noncommutative ring theory by extending the complex numbers into hypercomplex numbers. A rush of new systems followed within years. In 1844 Hamilton presented biquaternions, Cayley introduced octonions, and Grassmann introduced exterior algebras. James Cockle offered tessarines in 1848 and coquaternions in 1849, and William Kingdon Clifford added split-biquaternions in 1873. Cayley also brought in group algebras over the real and complex numbers in 1854 and square matrices across papers of 1855 and 1858. The flood of examples then demanded sorting. In an 1870 monograph, Benjamin Peirce classified more than 150 hypercomplex number systems of dimension below 6 and gave an explicit definition of an associative algebra, defining nilpotent and idempotent elements. Frobenius in 1878 and Charles Sanders Peirce in 1881 independently proved that the only finite-dimensional division algebras were the real numbers, the complex numbers, and the quaternions. In 1907 Wedderburn extended Cartan's results to an arbitrary field, yielding the Wedderburn principal theorem and the Artin-Wedderburn theorem. Commutative rings grew from a different soil. Gauss formulated the Gaussian integers in papers of 1828 and 1832, showing they form a unique factorization domain. In 1846 and 1847 Kummer introduced ideal numbers after Lamé's faulty 1847 proof of Fermat's Last Theorem assumed cyclotomic fields were unique factorization domains. Dedekind extended this in 1871, proving that every nonzero ideal in the integers of an algebraic number field is a unique product of prime ideals, work that founded algebraic number theory. The abstract ring itself came late. Abraham Fraenkel gave the first axiomatic definition in 1914, though his extra axioms on regular elements excluded even the ordinary integers. Masazo Sono's 1917 definition was the first equivalent to the modern one. Then in 1920 Emmy Noether, working with W. Schmeidler, defined left and right ideals, and her 1921 paper Idealtheorie in Ringbereichen analyzed ascending chain conditions on ideals. That work gave the term Noetherian ring, and Irving Kaplansky called it revolutionary, since results once tied to polynomial rings now followed from a single axiom.

• 5. Naming The Field

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  Richard Dedekind in 1871 reached for the German word Körper, meaning body or corpus, to name a set of real or complex numbers closed under the four arithmetic operations, suggesting an organically closed entity. The English term field arrived later, introduced by Moore in 1893. The structure had roots stretching back further. In 1801 Gauss introduced the integers mod p for a prime p, and Galois extended this in 1830 to finite fields. In 1881 Leopold Kronecker defined a domain of rationality, which in modern terms is a field of rational fractions. The first clear definition of an abstract field came from Heinrich Martin Weber in 1893, covering finite fields and the fields of algebraic number theory and geometry, though it omitted the associative law for multiplication. Steinitz pulled the threads together in 1910, defining fields with the modern axioms, classifying them by characteristic, and proving many theorems still seen today. His synthesis would soon feed back into the study of rings.

• 6. Structure Over Equations

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  The end of the 19th century and the start of the 20th brought a shift in how mathematics was done, driven by a demand for greater rigor. Mathematicians grew dissatisfied with merely establishing properties of concrete objects and turned toward general theory. Results about particular groups of permutations came to be seen as cases of theorems about an abstract group, and questions of structure and classification moved to the front. The algebraic study of general fields by Ernst Steinitz, and of commutative and general rings by David Hilbert, Emil Artin, and Emmy Noether, built on earlier ideas of Ernst Kummer, Leopold Kronecker, and Richard Dedekind, who had considered ideals in commutative rings. Georg Frobenius and Issai Schur added the representation theory of groups. These developments from the last quarter of the 19th century and the first quarter of the 20th were laid out systematically in Bartel van der Waerden's Moderne Algebra, a two-volume work published in 1930-1931. That book turned algebra away from the theory of equations and toward the theory of algebraic structures. Yet this ran almost opposite to the actual history, since textbooks like van der Waerden's open each chapter with a formal definition before offering concrete examples.

• 7. A Hierarchy Of Abstraction

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  Almost all systems studied in abstract algebra are sets, so the theorems of set theory apply to them. Add a single binary operation and a set becomes a magma. Pile on more constraints and the structure climbs a ladder: associativity yields semigroups, while identity and inverses yield groups. Examples with one binary operation run from magma and quasigroup through monoid, semigroup, and group. Examples with several operations include the ring, the field, the module, the vector space, and algebras over a field, along with Lie algebras, lattices, and Boolean algebra. This ladder carries a trade-off. A ring has two binary operations meeting certain axioms, and because a ring is a group over one of its operations, the theorems of group theory carry over when studying rings. More general structures usually admit fewer nontrivial theorems and fewer applications, while more specialized ones grow richer. There is, in short, a balance between generality and the depth of what can be proved. Algebraic structures and their homomorphisms also form mathematical categories, and category theory offers a unified framework for properties and constructions that recur across different structures.

• 8. Algebra In The Wider World

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  The Poincaré conjecture, proved in 2003, shows abstract algebra reaching far beyond its own borders. It asserts that the fundamental group of a manifold, which encodes information about connectedness, can decide whether a manifold is a sphere. Algebraic topology of this kind uses algebraic objects to study topologies. Andrew Wiles drew on algebraic number theory, the study of number rings that generalize the integers, to prove Fermat's Last Theorem. Physics leans on the subject just as heavily. Groups represent symmetry operations, and group theory can simplify differential equations, while in gauge theory the requirement of local symmetry can be used to derive the equations governing a system. The symmetries there are described by Lie groups, and studying Lie groups and Lie algebras reveals much about a physical system. The number of force carriers in a theory equals the dimension of the Lie algebra, and these bosons interact with the force they mediate when the Lie algebra is nonabelian, a direct line from a century of abstraction to the particles that carry the forces of nature.