In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically
algebra, abstract algebra or modern algebra is the study of
algebraic structures. Algebraic structures include
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
,
rings,
fields,
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
,
vector spaces,
lattices, and
algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from
elementary algebra, the use of
variables to represent numbers in computation and reasoning.
Algebraic structures, with their associated
homomorphisms, form
mathematical categories.
Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures.
Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''
variety of groups''.
History
Before the nineteenth century,
algebra meant the study of the solution of polynomial equations. 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
''Moderne Algebra'' is a two-volume German textbook on graduate abstract algebra by , originally based on lectures given by Emil Artin in 1926 and by from 1924 to 1928. The English translation of 1949–1950 had the title ''Modern algebra'', th ...
, 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.
Muhammad ibn Mūsā al-Khwārizmī
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persians, Persian polymath from Khwarazm, who produced vastly influential works in Mathematics ...
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
is restricted to
, in symbolical algebra all rules of operations hold with no restrictions. Using this Peacock could show laws such as
, by letting
in
. 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,
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 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 1976 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 , 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
Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.
Early life and education
Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Chri ...
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 In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natura ...
. Dedekind and Miller independently characterized
Hamiltonian group In group theory, a Dedekind group is a group ''G'' such that every subgroup of ''G'' is normal.
All abelian groups are Dedekind groups.
A non-abelian Dedekind group is called a Hamiltonian group.
The most familiar (and smallest) example of a Hamil ...
s 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 1904 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
Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician.
Cockle was born on 14 January 1819. He was the second son of James Cockle, a surgeon, of Great Oakley, Essex. Educated at Charterh ...
presented
tessarine
In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as
:(u,v)(w,z) = (u w - v z, u z ...
s in 1848 and
coquaternion
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.
After introduction i ...
s 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
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
or
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
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplica ...
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
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence ''x''3 ≡ ''p'' (mod ''q'') is solvable; the word "reciprocity" comes from the form o ...
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,
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 1971 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
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.
Formal statement
Dirichlet's principle states that, if the funct ...
, 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