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
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, 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
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
,
fields
Fields may refer to:
Music
* Fields (band), an indie rock band formed in 2006
* Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song b ...
,
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 space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s,
lattices, and
algebras over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition a ...
. 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
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entail ...
, the use of
variables to represent numbers in computation and reasoning.
Algebraic structures, with their associated
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s, 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
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of stu ...
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
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
of groups''.
History
Before the nineteenth century,
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
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 equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s. 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
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
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
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
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
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fu ...
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 Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...
originated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. Fully symbolic algebra did not appear until
François Viète
François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...
's 1591
New Algebra
New is an adjective referring to something recently made, discovered, or created.
New or NEW may refer to:
Music
* New, singer of K-pop group The Boyz
Albums and EPs
* ''New'' (album), by Paul McCartney, 2013
* ''New'' (EP), by Regurgitator, ...
, 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
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
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 In the history of mathematics, the principle of permanence, or law of the permanence of equivalent forms, was the idea that algebraic operations like addition and multiplication should behave consistently in every number system, especially when d ...
to justify his argument, but his reasoning suffered from the
problem of induction. For example,
holds for the nonnegative
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, but not for general
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s.
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
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ...
, the
multiplicative group of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to ''n'' from the set \ of ''n'' non-negative integers form a group under multiplication modulo ''n'', called the multiplicative group of integers modulo ''n''. Equivalently, the ele ...
, and the more general concepts of
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
s and
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s. Klein's 1872
Erlangen program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
studied geometry and led to
symmetry groups such as the
Euclidean group and the group of
projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
s. 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
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations o ...
, 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 abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
. In 1870 Kronecker defined an abstract binary operation that was closed, commutative, associative, and had the left
cancellation property
In mathematics, the notion of cancellative is a generalization of the notion of invertible.
An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that .
An ...
, similar to the modern laws for a finite
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
. Weber's 1882 definition of a group was a closed binary operation that was associative and had left and right cancellation.
Walther von Dyck
Walther Franz Anton von Dyck (6 December 1856 – 5 November 1934), born Dyck () and later ennobled, was a German mathematician. He is credited with being the first to define a mathematical group, in the modern sense in . He laid the foundations ...
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 Chris ...
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 natur ...
. 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 Hamilt ...
s and introduced the notion of the
commutator of two elements. Burnside, Frobenius, and Molien created the
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
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 number
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group represen ...
s, specifically
William Rowan Hamilton
Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irela ...
's
quaternions in 1843. Many other number systems followed shortly. In 1844, Hamilton presented
biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
s, Cayley introduced
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s, and Grassman introduced
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
s.
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 Charte ...
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 in ...
s in 1849.
William Kingdon Clifford introduced
split-biquaternion
In mathematics, a split-biquaternion is a hypercomplex number of the form
:q = w + xi + yj + zk
where ''w'', ''x'', ''y'', and ''z'' are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each coefficient ''w'', ''x' ...
s in 1873. In addition Cayley introduced
group algebras over the real and complex numbers in 1854 and
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
in two papers of 1855 and 1858.
Once there were sufficient examples, it remained to classify them. In an 1870 monograph,
Benjamin Peirce
Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philoso ...
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
In ring theory, a Peirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.
The Peirce decomposition for associative algebras was introduced by . A similar but more complicated Peirce decomp ...
. Frobenius in 1878 and
Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism".
Educated as a chemist and employed as a scientist for t ...
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 (mathematics), field ''K''. The ...
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 integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s and showed that they form a
unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
(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 of ...
law for the
Eisenstein integers. The study of
Fermat's last theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...
led to the
algebraic integer
In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s. In 1847,
Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity ...
thought he had proven FLT, but his proof was faulty as he assumed all the
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...
s were UFDs, yet as Kummer pointed out,
was not a UFD. In 1846 and 1847 Kummer introduced
ideal number In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings. An ideal in the rin ...
s 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 domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
s. Overall, Dedekind's work created the subject of
algebraic number theory.
In the 1850s, Riemann introduced the fundamental concept of a
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
. 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 functi ...
, 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 mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies ...
. In the 1860s and 1870s, Clebsch, Gordan, Brill, and especially
M. Noether studied
algebraic function In mathematics, an algebraic function is a function that can be defined
as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
s 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