Branch of mathematics
This article is about a branch of mathematics. For the Swedish band, see
Abstrakt Algebra
.
"Modern algebra" redirects here. For van der Waerden's book, see
Moderne Algebra
.
In
mathematics
, more specifically
algebra
,
abstract algebra
or
modern algebra
is the study of
algebraic structures
, which are
sets
with specific
operations
acting on their elements.
[1]
Algebraic structures include
groups
,
rings
,
fields
,
modules
,
vector spaces
,
lattices
, and
algebras over a field
. 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
, 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
.
Algebraic structures, with their associated
homomorphisms
, form
mathematical categories
.
Category theory
gives a unified framework to study 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
[
edit
]
Before the nineteenth century,
algebra
was defined as the study of
polynomials
.
Abstract algebra came into existence during the nineteenth century as more complex problems and solution methods developed. Concrete problems and examples came from number theory, geometry, analysis, and the solutions of
algebraic equations
. Most theories that are now recognized as parts of abstract algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. This unification occurred in the early decades of the 20th century and resulted in the formal
axiomatic
definitions of various
algebraic structures
such as groups, rings, and fields.
This historical development is almost the opposite of the treatment found in popular textbooks, such as van der Waerden's
Moderne Algebra
,
[4]
which start each chapter with a formal definition of a structure and then follow it with concrete examples.
Elementary algebra
[
edit
]
The study of polynomial equations or
algebraic equations
has a long history.
c.
1700 BC
, the Babylonians were able to solve quadratic equations specified as word problems. This word problem stage is classified as
rhetorical algebra
and was the dominant approach up to the 16th century.
Al-Khwarizmi
originated the word "algebra" in 830 AD, but his work was entirely rhetorical algebra. Fully symbolic algebra did not appear until
Francois Viete
's 1591
New Algebra
, and even this had some spelled out words that were given symbols in Descartes's 1637
La Geometrie
.
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.
[7]
However, European mathematicians, for the most part, resisted these concepts until the middle of the 19th century.
[8]
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
[
edit
]
Several areas of mathematics led to the study of groups. Lagrange's 1770 study of the solutions of the quintic equation led to the
Galois group of a polynomial
. Gauss's 1801 study of
Fermat's little theorem
led to the
ring of integers modulo n
, the
multiplicative group of integers modulo n
, and the more general concepts of
cyclic groups
and
abelian groups
. Klein's 1872
Erlangen program
studied geometry and led to
symmetry groups
such as the
Euclidean group
and the group of
projective transformations
. In 1874 Lie introduced the theory of
Lie groups
, aiming for "the Galois theory of differential equations". In 1876 Poincare and Klein introduced the group of
Mobius 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",
[11]
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
.
[13]
In 1870 Kronecker defined an abstract binary operation that was closed, commutative, associative, and had the left
cancellation property
,
[14]
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.
[19]
Otto Holder
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?Holder theorem
. Dedekind and Miller independently characterized
Hamiltonian groups
and introduced the notion of the
commutator
of two elements. Burnside, Frobenius, and Molien created the
representation theory
of finite groups at the end of the nineteenth century.
J. A. de Seguier'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
[
edit
]
Noncommutative ring theory began with extensions of the complex numbers to
hypercomplex numbers
, specifically
William Rowan Hamilton
's
quaternions
in 1843. Many other number systems followed shortly. In 1844, Hamilton presented
biquaternions
, Cayley introduced
octonions
, and Grassman introduced
exterior algebras
.
James Cockle
presented
tessarines
in 1848
[22]
and
coquaternions
in 1849.
[23]
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
and
Artin?Wedderburn theorem
.
For commutative rings, several areas together led to commutative ring theory.
In two papers in 1828 and 1832, Gauss formulated the
Gaussian integers
and showed that they form a
unique factorization domain
(UFD) and proved the
biquadratic reciprocity
law. Jacobi and Eisenstein at around the same time proved a
cubic reciprocity
law for the
Eisenstein integers
.
The study of
Fermat's last theorem
led to the
algebraic integers
. In 1847,
Gabriel Lame
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 1871 to show that every nonzero ideal in the domain of integers of an algebraic number field is a unique product of
prime ideals
, a precursor of the theory of
Dedekind domains
. Overall, Dedekind's work created the subject of
algebraic number theory
.
In the 1850s, Riemann introduced the fundamental concept of a
Riemann surface
. Riemann's methods relied on an assumption he called
Dirichlet's principle
,
which in 1870 was questioned by Weierstrass. Much later, in 1900, Hilbert justified Riemann's approach by developing the
direct method in the calculus of variations
.
[31]
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
, although Noether did not use this modern language. In 1882 Dedekind and Weber, in analogy with Dedekind's earlier work on algebraic number theory, created a theory of
algebraic function fields
which allowed the first rigorous definition of a Riemann surface and a rigorous proof of the
Riemann?Roch theorem
. Kronecker in the 1880s, Hilbert in 1890, Lasker in 1905, and Macauley in 1913 further investigated the ideals of polynomial rings implicit in
E. Noether
's work. Lasker proved a special case of the
Lasker-Noether theorem
, namely that every ideal in a polynomial ring is a finite intersection of
primary ideals
. Macauley proved the uniqueness of this decomposition.
Overall, this work led to the development of
algebraic geometry
.
In 1801 Gauss introduced
binary quadratic forms
over the integers and defined their
equivalence
. He further defined the
discriminant
of these forms, which is an
invariant of a binary form
. Between the 1860s and 1890s
invariant theory
developed and became a major field of algebra. Cayley, Sylvester, Gordan and others found the
Jacobian
and the
Hessian
for binary quartic forms and cubic forms.
In 1868 Gordan proved that the
graded algebra
of invariants of a binary form over the complex numbers was finitely generated, i.e., has a basis.
[34]
Hilbert wrote a thesis on invariants in 1885 and in 1890 showed that any form of any degree or number of variables has a basis. He extended this further in 1890 to
Hilbert's basis theorem
.
Once these theories had been developed, it was still several decades until an abstract ring concept emerged. The first axiomatic definition was given by
Abraham Fraenkel
in 1914.
His definition was mainly the standard axioms: a set with two operations addition, which forms a group (not necessarily commutative), and multiplication, which is associative, distributes over addition, and has an identity element.
[36]
In addition, he had two axioms on "regular elements" inspired by work on the
p-adic numbers
, which excluded now-common rings such as the ring of integers. These allowed Fraenkel to prove that addition was commutative.
[37]
Fraenkel's work aimed to transfer Steinitz's 1910 definition of fields over to rings, but it was not connected with the existing work on concrete systems. Masazo Sono's 1917 definition was the first equivalent to the present one.
In 1920,
Emmy Noether
, in collaboration with W. Schmeidler, published a paper about the
theory of ideals
in which they defined
left and right ideals
in a
ring
. The following year she published a landmark paper called
Idealtheorie in Ringbereichen
(
Ideal theory in rings'
), analyzing
ascending chain conditions
with regard to (mathematical) ideals. The publication gave rise to the term "
Noetherian ring
", and several other mathematical objects being called
Noetherian
.
[40]
Noted algebraist
Irving Kaplansky
called this work "revolutionary";
results which seemed inextricably connected to properties of polynomial rings were shown to follow from a single axiom.
Artin, inspired by Noether's work, came up with the
descending chain condition
. These definitions marked the birth of abstract ring theory.
Early field theory
[
edit
]
In 1801 Gauss introduced the
integers mod p
, where p is a prime number. Galois extended this in 1830 to
finite fields
with
elements.
In 1871
Richard Dedekind
introduced, for a set of real or complex numbers that is closed under the four arithmetic operations,
the
German
word
Korper
, which means "body" or "corpus" (to suggest an organically closed entity). The English term "field" was introduced by Moore in 1893.
[45]
In 1881
Leopold Kronecker
defined what he called a
domain of rationality
, which is a field of
rational fractions
in modern terms.
The first clear definition of an abstract field was due to
Heinrich Martin Weber
in 1893. It was missing the associative law for multiplication, but covered finite fields and the fields of algebraic number theory and algebraic geometry.
In 1910 Steinitz synthesized the knowledge of abstract field theory accumulated so far. He axiomatically defined fields with the modern definition, classified them by their
characteristic
, and proved many theorems commonly seen today.
Other major areas
[
edit
]
Modern algebra
[
edit
]
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
in mathematics. Initially, the assumptions in classical
algebra
, on which the whole of mathematics (and major parts of the
natural sciences
) depend, took the form of
axiomatic systems
. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain
algebraic structures
began to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of an
abstract group
. Questions of structure and classification of various mathematical objects came to forefront.
[
citation needed
]
These processes were occurring throughout all of mathematics, but became especially pronounced in algebra. Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such as
groups
,
rings
, and
fields
. Hence such things as
group theory
and
ring theory
took their places in
pure mathematics
. The algebraic investigations of general fields by
Ernst Steinitz
and of commutative and then general rings by
David Hilbert
,
Emil Artin
and
Emmy Noether
, building on the work of
Ernst Kummer
,
Leopold Kronecker
and
Richard Dedekind
, who had considered ideals in commutative rings, and of
Georg Frobenius
and
Issai Schur
, concerning
representation theory
of groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of 20th century were systematically exposed in
Bartel van der Waerden
's
Moderne Algebra
, the two-volume
monograph
published in 1930?1931 that reoriented the idea of algebra from
the theory of equations
to
the
theory of algebraic structures
.
[
citation needed
]
Basic concepts
[
edit
]
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
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
semigroups
); 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
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
theorems and fewer applications.
[
citation needed
]
Examples of algebraic structures with a single
binary operation
are:
Examples involving several operations include:
Branches of abstract algebra
[
edit
]
Group theory
[
edit
]
A group is a set
together with a "group product", a binary operation
. The group satisfies the following defining axioms (c.f.
Group (mathematics) § Definition
):
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
[
edit
]
A ring is a set
with two
binary operations
, addition:
and multiplication:
satisfying the following
axioms
.
Applications
[
edit
]
Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance,
algebraic topology
uses algebraic objects to study topologies. The
Poincare conjecture
, proved in 2003, asserts that the
fundamental group
of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not.
Algebraic number theory
studies various number
rings
that generalize the set of integers. Using tools of algebraic number theory,
Andrew Wiles
proved
Fermat's Last Theorem
.
[
citation needed
]
In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In
gauge theory
, the requirement of
local symmetry
can be used to deduce the equations describing a system. The groups that describe those symmetries are
Lie groups
, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of
force carriers
in a theory is equal to the dimension of the Lie algebra, and these
bosons
interact with the force they mediate if the Lie algebra is nonabelian.
[50]
See also
[
edit
]
References
[
edit
]
- ^
Finston, David R.; Morandi, Patrick J. (29 August 2014).
Abstract Algebra: Structure and Application
. Springer. p. 58.
ISBN
978-3-319-04498-9
.
Much of our study of abstract algebra involves an analysis of structures and their operations
- ^
van der Waerden, Bartel Leendert (1949).
Modern Algebra. Vol I
. Translated by Blum, Fred. New York, N. Y.: Frederick Ungar Publishing Co.
MR
0029363
.
- ^
Euler, Leonard (1748).
Introductio in Analysin Infinitorum
[
Introduction to the Analysis of the Infinite
] (in Latin). Vol. 1. Lucerne, Switzerland: Marc Michel Bosquet & Co. p. 104.
- ^
Martinez, Alberto (2014).
Negative Math
. Princeton University Press. pp. 80?109.
- ^
O'Connor, John J.;
Robertson, Edmund F.
,
"The abstract group concept"
,
MacTutor History of Mathematics Archive
,
University of St Andrews
- ^
Cayley, A. (1854).
"On the theory of groups, as depending on the symbolic equation θ
n
= 1"
.
Philosophical Magazine
. 4th series.
7
(42): 40?47.
doi
:
10.1080/14786445408647421
.
- ^
Kronecker, Leopold (1895). "Auseinandeesetzung einiger eigenschaften der klassenanzahl idealer complexer zahlen" [An exposition of some properties of the class number of ideal complex numbers]. In Hensel, Kurt (ed.).
Leopold Kronecker's werke : Herausgegeben auf veranlassung der Koniglich preussischen akademie der wissenschaften
. Leipzig; Berlin: B.G. Teubner. p. 275.
- ^
Frobenius, G. (April 2008) [1887].
"Neuer Beweis des Sylowschen Satzes"
[New Proof of Sylow's Theorem]
(PDF)
.
Journal fur die reine und angewandte Mathematik
(in German).
1887
(100). Translated by Gutfraind, Sasha: 179?181.
doi
:
10.1515/crll.1887.100.179
.
S2CID
117970003
.
- ^
Cockle, James (1848).
"On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra"
.
The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science
.
33
. Taylor & Francis: 435?9.
doi
:
10.1080/14786444808646139
.
- ^
Cockle, James (1849).
"On Systems of Algebra involving more than one Imaginary"
.
The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science
.
35
. Taylor & Francis: 434?7.
doi
:
10.1080/14786444908646384
.
- ^
Monna 1975
, pp. 55?56, citing
Hilbert, David (1905), "Uber das Dirichletsche Prinzip",
Journal fur die reine und angewandte Mathematik
(in German), vol. 129, pp. 63?67
- ^
Gordan, Paul (1868),
"Beweis, dass jede Covariante und Invariante einer binaren Form eine ganze Funktion mit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist"
,
Journal fur die reine und angewandte Mathematik
, vol. 1868, no. 69, pp. 323?354,
doi
:
10.1515/crll.1868.69.323
,
S2CID
120689164
- ^
Frankel, A. (1914) "Uber die Teiler der Null und die Zerlegung von Ringen". J. Reine Angew. Math. 145: 139?176
- ^
Corry, Leo (January 2000).
"The origins of the definition of abstract rings"
.
Modern Logic
.
8
(1?2): 5?27.
ISSN
1047-5982
.
- ^
Dick, Auguste
(1981),
Emmy Noether: 1882?1935
, translated by Blocher, H. I.,
Birkhauser
,
ISBN
3-7643-3019-8
, p. 44?45.
- ^
"
Earliest Known Uses of Some of the Words of Mathematics (F)
"
.
- ^
Hart, Roger (2011).
The Chinese roots of linear algebra
. Baltimore, MD: Johns Hopkins University Press.
ISBN
978-0-8018-9958-4
.
OCLC
794700410
.
- ^
Schumm, Bruce (2004),
Deep Down Things
, Baltimore: Johns Hopkins University Press,
ISBN
0-8018-7971-X
Bibliography
[
edit
]
- Gray, Jeremy (2018).
A history of abstract algebra: from algebraic equations to modern algebra
. Springer Undergraduate Mathematics Series. Cham, Switzerland.
doi
:
10.1007/978-3-319-94773-0
.
ISBN
978-3-319-94773-0
.
S2CID
125927783
.
{{
cite book
}}
: CS1 maint: location missing publisher (
link
)
- Kimberling, Clark
(1981). "Emmy Noether and Her Influence". In Brewer, James W; Smith, Martha K (eds.).
Emmy Noether: A Tribute to Her Life and Work
.
Marcel Dekker
. pp. 3?61.
- Kleiner, Israel (2007). Kleiner, Israel (ed.).
A history of abstract algebra
. Boston, Mass.: Birkhauser.
doi
:
10.1007/978-0-8176-4685-1
.
ISBN
978-0-8176-4685-1
.
- Monna, A. F. (1975),
Dirichlet's principle: A mathematical comedy of errors and its influence on the development of analysis
, Oosthoek, Scheltema & Holkema,
ISBN
978-9031301751
Further reading
[
edit
]
- Allenby, R. B. J. T. (1991),
Rings, Fields and Groups
, Butterworth-Heinemann,
ISBN
978-0-340-54440-2
- Artin, Michael
(1991),
Algebra
,
Prentice Hall
,
ISBN
978-0-89871-510-1
- Burris, Stanley N.; Sankappanavar, H. P. (1999) [1981],
A Course in Universal Algebra
- Gilbert, Jimmie; Gilbert, Linda (2005),
Elements of Modern Algebra
, Thomson Brooks/Cole,
ISBN
978-0-534-40264-8
- Lang, Serge
(2002),
Algebra
,
Graduate Texts in Mathematics
, vol. 211 (Revised third ed.), New York: Springer-Verlag,
ISBN
978-0-387-95385-4
,
MR
1878556
- Sethuraman, B. A. (1996),
Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility
, Berlin, New York:
Springer-Verlag
,
ISBN
978-0-387-94848-5
- Whitehead, C. (2002),
Guide to Abstract Algebra
(2nd ed.), Houndmills: Palgrave,
ISBN
978-0-333-79447-0
- W. Keith Nicholson (2012)
Introduction to Abstract Algebra
, 4th edition,
John Wiley & Sons
ISBN
978-1-118-13535-8
.
- John R. Durbin (1992)
Modern Algebra : an introduction
, John Wiley & Sons
External links
[
edit
]
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