Before the nineteenth century,
algebra was defined as the study of
polynomials. Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed. Concrete problems and examples came from number theory, geometry, analysis, and the solutions of
algebraic equations. Most theories that are now recognized as parts of abstract algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. This unification occurred in the early decades of the 20th century and resulted in the formal
axiomatic definitions of various
algebraic structures such as groups, rings, and fields. This historical development is almost the opposite of the treatment found in popular textbooks, such as van der Waerden's
Moderne Algebra, which start each chapter with a formal definition of a structure and then follow it with concrete examples.
Elementary algebra The study of polynomial equations or
algebraic equations has a long history. Circa 1700 BC, the Babylonians were able to solve quadratic equations specified as word problems. This word problem stage is classified as
rhetorical algebra and was the dominant approach up to the 16th century.
Al-Khwarizmi originated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. Fully symbolic algebra did not appear until
François Viète's 1591
New Algebra, and even this had some spelled out words that were given symbols in Descartes's 1637
La Géométrie. The formal study of solving symbolic equations led
Leonhard Euler to accept what were then considered "nonsense" roots such as
negative numbers and
imaginary numbers, in the late 18th century. However, European mathematicians, for the most part, resisted these concepts until the middle of the 19th century.
George Peacock's 1830
Treatise of Algebra was the first attempt to place algebra on a strictly symbolic basis. He distinguished a new
symbolical algebra, distinct from the old
arithmetical algebra. Whereas in arithmetical algebra a - b is restricted to a \geq b, in symbolical algebra all rules of operations hold with no restrictions. Using this Peacock could show laws such as (-a)(-b) = ab, by letting d=0,c=0 in (d - b)(c - a)=dc + ba - ad - bc. Peacock used what he termed the
principle of the permanence of equivalent forms to justify his argument, but his reasoning suffered from the
problem of induction. For example, \sqrt{a} \sqrt{b} = \sqrt{ab} holds for the nonnegative
real numbers, but not for general
complex numbers.
Early group theory Several areas of mathematics led to the study of groups. Lagrange's 1770 study of the solutions of the quintic equation led to the
Galois group of a polynomial. Gauss's 1801 study of
Fermat's little theorem led to the
ring of integers modulo n, the
multiplicative group of integers modulo n, and the more general concepts of
cyclic groups and
abelian groups. Klein's 1872
Erlangen program studied geometry and led to
symmetry groups such as the
Euclidean group and the group of
projective transformations. In 1874 Lie introduced the theory of
Lie groups, aiming for "the Galois theory of differential equations". In 1876 Poincaré and Klein introduced the group of
Möbius transformations, and its subgroups such as the
modular group and
Fuchsian group, based on work on automorphic functions in analysis. The abstract concept of group emerged slowly over the middle of the nineteenth century. Galois in 1832 was the first to use the term "group", signifying a collection of permutations closed under composition.
Arthur Cayley's 1854 paper
On the theory of groups defined a group as a set with an associative composition operation and the identity 1, today called a
monoid. In 1870 Kronecker defined an abstract binary operation that was closed, commutative, associative, and had the left
cancellation property b\neq c \to a\cdot b\neq a\cdot c, similar to the modern laws for a finite
abelian group. Weber's 1882 definition of a group was a closed binary operation that was associative and had left and right cancellation.
Walther von Dyck in 1882 was the first to require inverse elements as part of the definition of a group. Once this abstract group concept emerged, results were reformulated in this abstract setting. For example,
Sylow's theorem was reproven by Frobenius in 1887 directly from the laws of a finite group, although Frobenius remarked that the theorem followed from Cauchy's theorem on permutation groups and the fact that every finite group is a subgroup of a permutation group.
Otto Hölder was particularly prolific in this area, defining quotient groups in 1889, group automorphisms in 1893, as well as simple groups. He also completed the
Jordan–Hölder theorem. Dedekind and Miller independently characterized
Hamiltonian groups and introduced the notion of the
commutator of two elements. Burnside, Frobenius, and Molien created the
representation theory of finite groups at the end of the nineteenth century. J. A. de Séguier's 1905 monograph
Elements of the Theory of Abstract Groups presented many of these results in an abstract, general form, relegating "concrete" groups to an appendix, although it was limited to finite groups. The first monograph on both finite and infinite abstract groups was O. K. Schmidt's 1916
Abstract Theory of Groups.
Early ring theory Noncommutative ring theory began with extensions of the complex numbers to
hypercomplex numbers, specifically
William Rowan Hamilton's
quaternions in 1843. Many other number systems followed shortly. In 1844, Hamilton presented
biquaternions, Cayley introduced
octonions, and Grassman introduced
exterior algebras.
James Cockle presented
tessarines in 1848 and
coquaternions in 1849.
William Kingdon Clifford introduced
split-biquaternions in 1873. In addition Cayley introduced
group algebras over the real and complex numbers in 1854 and
square matrices in two papers of 1855 and 1858. Once there were sufficient examples, it remained to classify them. In an 1870 monograph,
Benjamin Peirce classified the more than 150 hypercomplex number systems of dimension below 6, and gave an explicit definition of an
associative algebra. He defined nilpotent and idempotent elements and proved that any algebra contains one or the other. He also defined the
Peirce decomposition. Frobenius in 1878 and
Charles Sanders Peirce in 1881 independently proved that the only finite-dimensional division algebras over \mathbb{R} were the real numbers, the complex numbers, and the quaternions. In the 1880s Killing and Cartan showed that semisimple
Lie algebras could be decomposed into simple ones, and classified all simple Lie algebras. Inspired by this, in the 1890s Cartan, Frobenius, and Molien proved (independently) that a finite-dimensional associative algebra over \mathbb{R} or \mathbb{C} uniquely decomposes into the
direct sums of a nilpotent algebra and a semisimple algebra that is the product of some number of
simple algebras, square matrices over division algebras. Cartan was the first to define concepts such as direct sum and simple algebra, and these concepts proved quite influential. In 1907 Wedderburn extended Cartan's results to an arbitrary field, in what are now called the
Wedderburn principal theorem and
Artin–Wedderburn theorem. For commutative rings, several areas together led to commutative ring theory. In two papers in 1828 and 1832, Gauss formulated the
Gaussian integers and showed that they form a
unique factorization domain (UFD) and proved the
biquadratic reciprocity law. Jacobi and Eisenstein at around the same time proved a
cubic reciprocity law for the
Eisenstein integers. The study of
Fermat's Last Theorem led to the
algebraic integers. In 1847,
Gabriel Lamé thought he had proven FLT, but his proof was faulty as he assumed all the
cyclotomic fields were UFDs, yet as Kummer pointed out, \mathbb{Q}(\zeta_{23})) was not a UFD. In 1846 and 1847 Kummer introduced
ideal numbers and proved unique factorization into ideal primes for cyclotomic fields. Dedekind extended this in 1871 to show that every nonzero ideal in the domain of integers of an algebraic number field is a unique product of
prime ideals, a precursor of the theory of
Dedekind domains. Overall, Dedekind's work created the subject of
algebraic number theory. In the 1850s, Riemann introduced the fundamental concept of a
Riemann surface. Riemann's methods relied on an assumption he called
Dirichlet's principle, which in 1870 was questioned by Weierstrass. Much later, in 1900, Hilbert justified Riemann's approach by developing the
direct method in the calculus of variations. In the 1860s and 1870s, Clebsch, Gordan, Brill, and especially
M. Noether studied
algebraic functions and curves. In particular, Noether studied what conditions were required for a polynomial to be an element of the ideal generated by two algebraic curves in the polynomial ring \mathbb{R}[x, y], although Noether did not use this modern language. In 1882 Dedekind and Weber, in analogy with Dedekind's earlier work on algebraic number theory, created a theory of
algebraic function fields which allowed the first rigorous definition of a Riemann surface and a rigorous proof of the
Riemann–Roch theorem. Kronecker in the 1880s, Hilbert in 1890, Lasker in 1905, and
Macaulay in 1913 further investigated the ideals of polynomial rings implicit in
E. Noether's work. Lasker proved a special case of the
Lasker-Noether theorem, namely that every ideal in a polynomial ring is a finite intersection of
primary ideals. Macaulay proved the uniqueness of this decomposition. Overall, this work led to the development of
algebraic geometry. In 1801 Gauss introduced
binary quadratic forms over the integers and defined their
equivalence. He further defined the
discriminant of these forms, which is an
invariant of a binary form. Between the 1860s and 1890s
invariant theory developed and became a major field of algebra. Cayley, Sylvester, Gordan and others found the
Jacobian and the
Hessian for binary quartic forms and cubic forms. In 1868 Gordan proved that the
graded algebra of invariants of a binary form over the complex numbers was finitely generated, i.e., has a basis. Hilbert wrote a thesis on invariants in 1885 and in 1890 showed that any form of any degree or number of variables has a basis. He extended this further in 1890 to
Hilbert's basis theorem. Once these theories had been developed, it was still several decades until an abstract ring concept emerged. The first axiomatic definition was given by
Abraham Fraenkel in 1914. His definition was mainly the standard axioms: a set with two operations addition, which forms a group (not necessarily commutative), and multiplication, which is associative, distributes over addition, and has an identity element. In addition, he had two axioms on "regular elements" inspired by work on the
p-adic numbers, which excluded now-common rings such as the ring of integers. These allowed Fraenkel to prove that addition was commutative. Fraenkel's work aimed to transfer Steinitz's 1910 definition of fields over to rings, but it was not connected with the existing work on concrete systems. Masazo Sono's 1917 definition was the first equivalent to the present one. In 1920,
Emmy Noether, in collaboration with W. Schmeidler, published a paper about the
theory of ideals in which they defined
left and right ideals in a
ring. The following year she published a landmark paper called
Idealtheorie in Ringbereichen (
Ideal theory in rings'), analyzing
ascending chain conditions with regard to (mathematical) ideals. The publication gave rise to the term "
Noetherian ring", and several other mathematical objects being called
Noetherian. Noted algebraist
Irving Kaplansky called this work "revolutionary"; results which seemed inextricably connected to properties of polynomial rings were shown to follow from a single axiom. Artin, inspired by Noether's work, came up with the
descending chain condition. These definitions marked the birth of abstract ring theory.
Early field theory In 1801 Gauss introduced the
integers mod p, where p is a prime number. Galois extended this in 1830 to
finite fields with p^n elements. In 1871
Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the
German word
Körper, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore in 1893. In 1881
Leopold Kronecker defined what he called a
domain of rationality, which is a field of
rational fractions in modern terms. The first clear definition of an abstract field was due to
Heinrich Martin Weber in 1893. It was missing the associative law for multiplication, but covered finite fields and the fields of algebraic number theory and algebraic geometry. In 1910 Steinitz synthesized the knowledge of abstract field theory accumulated so far. He axiomatically defined fields with the modern definition, classified them by their
characteristic, and proved many theorems commonly seen today.
Other major areas • Solving of
systems of linear equations, which led to
linear algebra Modern algebra The end of the 19th and the beginning of the 20th century saw a change in the methodology of mathematics. Abstract algebra emerged around the start of the 20th century, under the name
modern algebra. Its study was part of the drive for more
intellectual rigor in mathematics. Initially, the assumptions in classical
algebra, on which the whole of mathematics (and major parts of the
natural sciences) depend, took the form of
axiomatic systems. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain
algebraic structures began to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of an
abstract group. Questions of structure and classification of various mathematical objects came to the forefront. These processes were occurring throughout all of mathematics but became especially noticeable in algebra. Formal definitions through primitive operations and axioms were proposed for many basic algebraic structures, such as
groups,
rings, and
fields. Hence such things as
group theory and
ring theory took their places in
pure mathematics. The algebraic investigations of general fields by
Ernst Steinitz and of commutative and then general rings by
David Hilbert,
Emil Artin and
Emmy Noether, building on the work of
Ernst Kummer,
Leopold Kronecker and
Richard Dedekind, who had considered ideals in commutative rings, and of
Georg Frobenius and
Issai Schur, concerning
representation theory of groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of the 20th century were systematically exposed in
Bartel van der Waerden's
Moderne Algebra, the two-volume
monograph published in 1930–1931 that reoriented the idea of algebra from
the theory of equations to
the theory of algebraic structures. ==Basic concepts==