In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more specifically
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, abstract algebra or modern algebra is the study of
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
s, which are
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
s with specific
operations acting on their elements. Algebraic structures include
groups,
rings,
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 by ...
,
modules,
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 ...
. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from
elementary algebra
Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics ...
, the use of
variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in
pedagogy
Pedagogy (), most commonly understood as the approach to teaching, is the theory and practice of learning, and how this process influences, and is influenced by, the social, political, and psychological development of learners. Pedagogy, taken ...
.
Algebraic structures, with their associated
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s, form
mathematical categories.
Category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
gives a unified framework to study 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 in general, not specific types of algebraic structures.
For instance, rather than considering groups or rings as the object of stud ...
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
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
was defined as the study of
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s. 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 example, x^5-3x+1=0 is an algebraic equati ...
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
axiom
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 ...
atic definitions of various
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
s 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'', tho ...
'', 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 example, x^5-3x+1=0 is an 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 fun ...
and was the dominant approach up to the 16th century.
Al-Khwarizmi
Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
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 (; 1540 – 23 February 1603), known in Latin as Franciscus Vieta, was a French people, French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as par ...
's 1591
New Algebra
New or NEW may refer to:
Music
* New, singer of K-pop group The Boyz
* ''New'' (album), by Paul McCartney, 2013
** "New" (Paul McCartney song), 2013
* ''New'' (EP), by Regurgitator, 1995
* "New" (Daya song), 2017
* "New" (No Doubt song), 1 ...
, and even this had some spelled out words that were given symbols in Descartes's 1637 ''
La Géométrie
''La Géométrie'' () was published in 1637 as an appendix to ''Discours de la méthode'' ('' Discourse on the Method''), written by René Descartes. In the ''Discourse'', Descartes presents his method for obtaining clarity on any subject. ''La ...
''. The formal study of solving symbolic equations led
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
to accept what were then considered "nonsense" roots such as
negative number
In mathematics, a negative number is the opposite (mathematics), opposite of a positive real number. Equivalently, a negative number is a real number that is inequality (mathematics), less than 0, zero. Negative numbers are often used to represe ...
s and
imaginary number
An imaginary number is the product of a real number and the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square (algebra), square of an im ...
s, 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
The problem of induction is a philosophical problem that questions the rationality of predictions about unobserved things based on previous observations. These inferences from the observed to the unobserved are known as "inductive inferences" ...
. 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 for ...
s.
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
In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as
a^p \equiv a \pmod p.
For example, if and , t ...
led to the
ring of integers modulo n, 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 el ...
, and the more general concepts of
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
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 commu ...
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 na ...
studied geometry and led to
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
s such as the
Euclidean group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformati ...
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 group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s, aiming for "the Galois theory of differential equations". In 1876 Poincaré and Klein introduced the group of
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying .
Geometrically ...
s, and its subgroups such as the
modular group
In mathematics, the modular group is the projective special linear group \operatorname(2,\mathbb Z) of 2\times 2 matrices with integer coefficients and determinant 1, such that the matrices A and -A are identified. The modular group acts on ...
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
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years.
He ...
'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 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 .
Monoids are semigroups with identity ...
. 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
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 commu ...
. 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 foundation ...
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 Christ ...
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 na ...
. Dedekind and Miller independently characterized
Hamiltonian groups and introduced the notion of the
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
of two elements. Burnside, Frobenius, and Molien created the
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
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 number
In mathematics, hypercomplex number is a traditional term for an element (mathematics), element of a finite-dimensional Algebra over a field#Unital algebra, unital algebra over a field, algebra over the field (mathematics), field of real numbers. ...
s, specifically
William Rowan Hamilton
Sir William Rowan Hamilton (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...
's
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s 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 cor ...
s, Cayley introduced
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...
s, and Grassman introduced
exterior algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
s.
James Cockle presented
tessarines in 1848 and
coquaternions in 1849.
William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was a British mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...
introduced
split-biquaternion
In mathematics, a split-biquaternion is a hypercomplex number of the form
: q = w + x\mathrm + y\mathrm + z\mathrm ,
where ''w'', ''x'', ''y'', and ''z'' are split-complex numbers and i, j, and k multiply as in the quaternion group. Since each co ...
s 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
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
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
. He defined nilpotent and idempotent elements and proved that any algebra contains one or the other. He also defined the
Peirce decomposition Peirce may refer to:
* Charles Sanders Peirce (1839–1914), American philosopher, founder of pragmatism
Schools
* Peirce College, Philadelphia, formerly known as Peirce College of Business, Peirce Junior College and Peirce School of Business Admi ...
. Frobenius in 1878 and
Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss (philosopher), Paul ...
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 algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s 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 algebra In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a sim ...
s, 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 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 ...
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 integer
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
: z = a + b\omega ,
where and are integers and
: \omega = \frac ...
s. 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 number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
led to the
algebraic integer
In algebraic number theory, an algebraic integer is a complex number that 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 thought he had proven FLT, but his proof was faulty as he assumed all the
cyclotomic field
In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory ...
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 r ...
s 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 ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...
s, a precursor of the theory of
Dedekind domain
In mathematics, 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 un ...
s. Overall, Dedekind's work created the subject of
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
.
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 vers ...
. 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 function
In mathematics, an algebraic function is a function that can be defined
as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operati ...
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
, y
The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math>, 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 field
In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebrai ...
s which allowed the first rigorous definition of a Riemann surface and a rigorous proof of the
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It re ...
. 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 ideal
In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y'n'' is also an element of ''Q'', for some ''n'' > 0. ...
s. Macaulay proved the uniqueness of this decomposition. Overall, this work led to the development of
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
.
In 1801 Gauss introduced
binary quadratic form
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables
: q(x,y)=ax^2+bxy+cy^2, \,
where ''a'', ''b'', ''c'' are the coefficients. When the coefficients can be arbitrary complex numbers, most results ar ...
s over the integers and defined their
equivalence. He further defined the
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...
of these forms, which is an
invariant of a binary form. Between the 1860s and 1890s
invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descr ...
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
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, ...
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
In mathematics Hilbert's basis theorem asserts that every ideal (ring theory), ideal of a polynomial ring over a field (mathematics), field has a finite generating set of an ideal, generating set (a finite ''basis'' in Hilbert's terminology).
In ...
.
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
Abraham Fraenkel (; 17 February, 1891 – 15 October, 1965) was a German-born Israeli mathematician. He was an early Zionist and the first Dean of Mathematics at the Hebrew University of Jerusalem. He is known for his contributions to axiomatic ...
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 number
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infin ...
s, 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
Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
, 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 condition
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
s with regard to (mathematical) ideals. The publication gave rise to the term "
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
", and several other mathematical objects being called ''
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
''. Noted algebraist
Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St And ...
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
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings. These conditions played an important r ...
. 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 field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s with
elements. In 1871
Richard Dedekind
Julius Wilhelm Richard Dedekind (; ; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. H ...
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
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker
as having said, ...
defined what he called a ''domain of rationality'', which is a field of
rational fraction
In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmetic fractions.
A ration ...
s 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
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables.
For example,
: \begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in ...
, which led to
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
Modern algebra
The end of the 19th and the beginning of the 20th century saw a shift 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
Rigour (British English) or rigor (American English; American and British English spelling differences#-our, -or, see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such ...
in mathematics. Initially, the assumptions in classical
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, on which the whole of mathematics (and major parts of the
natural sciences
Natural science or empirical science is one of the branches of science concerned with the description, understanding and prediction of natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer ...
) depend, took the form of
axiomatic system
In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemmas or theorems. A proof within an axiom system is a sequence of deductive steps that establishes ...
s. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
s 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 pronounced in algebra. Formal definitions through primitive operations and axioms were proposed for many basic algebraic structures, such as
groups,
rings, 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 by ...
. Hence such things as
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
and
ring theory took their places in
pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications ...
. The algebraic investigations of general fields by
Ernst Steinitz and of commutative and then general rings by
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ...
,
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...
and
Emmy Noether
Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
, building on the work of
Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a '' gymnasium'', the German equivalent of h ...
,
Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker
as having said, ...
and
Richard Dedekind
Julius Wilhelm Richard Dedekind (; ; 6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. H ...
, who had considered ideals in commutative rings, and of
Georg Frobenius and
Issai Schur
Issai Schur (10 January 1875 – 10 January 1941) was a Russian mathematician who worked in Germany for most of his life. He studied at the Humboldt University of Berlin, University of Berlin. He obtained his doctorate in 1901, became lecturer i ...
, concerning
representation theory
Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
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
''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'', tho ...
'', the two-volume
monograph
A monograph is generally a long-form work on one (usually scholarly) subject, or one aspect of a subject, typically created by a single author or artist (or, sometimes, by two or more authors). Traditionally it is in written form and published a ...
published in 1930–1931 that reoriented the idea of algebra from ''the theory of equations'' to ''the'' ''theory of algebraic structures''.
Basic concepts
By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied are
sets, to which the theorems of
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
apply. Those sets that have a certain binary operation defined on them form
magmas, to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to form
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
s); identity, and inverses (to form
groups); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
may be used when studying
rings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. In general there is a balance between the amount of generality and the richness of the theory: more general structures have usually fewer
nontrivial
In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group (mathematics), group, topological space). The n ...
theorems and fewer applications.

Examples of algebraic structures with a single
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a binary operation ...
are:
*
Magma
Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma (sometimes colloquially but incorrectly referred to as ''lava'') is found beneath the surface of the Earth, and evidence of magmatism has also ...
*
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element pro ...
*
Monoid
In abstract algebra, 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 .
Monoids are semigroups with identity ...
*
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
*
Group
Examples involving several operations include:
*
Ring
*
Field
*
Module
*
Vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
*
Algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set to ...
*
Associative algebra
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
*
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
*
Lattice
*
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
Branches of abstract algebra
Group theory
A group is a set
together with a "group product", a binary operation
. The group satisfies the following defining axioms (cf. ):
Identity: there exists an element
such that, for each element
in
, it holds that
.
Inverse: for each element
of
, there exists an element
so that
.
Associativity: for each triplet of elements
in
, it holds that
.
Ring theory
A ring is a set
with two
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a binary operation ...
s, addition:
and multiplication:
satisfying the following
axiom
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 ...
s.
*
is a
commutative group under addition.
*
is a
monoid
In abstract algebra, 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 .
Monoids are semigroups with identity ...
under multiplication.
*Multiplication is
distributive with respect to addition.
Applications
Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance,
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
uses algebraic objects to study topologies. The
Poincaré conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
Originally conjectured b ...
, proved in 2003, asserts that the
fundamental group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not.
Algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
studies various number
rings that generalize the set of integers. Using tools of algebraic number theory,
Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for Wiles's proof of Fermat's Last Theorem, proving Ferma ...
proved
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 number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
.
In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
, the requirement of
local symmetry
The symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.
A family of particular transformations may be ''continuous'' (such ...
can be used to deduce the equations describing a system. The groups that describe those symmetries are
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of
force carrier
In quantum field theory, a force carrier is a type of particle that gives rise to forces between other particles. They serve as the quanta of a particular kind of physical field. Force carriers are also known as messenger particles, intermedia ...
s in a theory is equal to the dimension of the Lie algebra, and these
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
s interact with the force they mediate if the Lie algebra is nonabelian.
See also
*
Coding theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and computer data storage, data sto ...
*
Group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
*
List of publications in abstract algebra
This is a list of publications in mathematics, organized by field.
Some reasons a particular publication might be regarded as important:
*Topic creator – A publication that created a new topic
*Breakthrough – A publication that cha ...
References
Bibliography
*
*
*
*
Further reading
*
*
*
*
*
*
*
* W. Keith Nicholson (2012) ''Introduction to Abstract Algebra'', 4th edition,
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and pr ...
.
* John R. Durbin (1992) ''Modern Algebra : an introduction'', John Wiley & Sons
External links
* Charles C. Pinter (1990)
982A Book of Abstract Algebra', second edition, from
University of Maryland
The University of Maryland, College Park (University of Maryland, UMD, or simply Maryland) is a public land-grant research university in College Park, Maryland, United States. Founded in 1856, UMD is the flagship institution of the Univ ...
{{DEFAULTSORT:Abstract Algebra