In algebra, which is a broad division of mathematics, abstract algebra is a common name for the
sub-area that studies algebraic structures in their own right. Such structures include
groups, rings, fields, modules, vector spaces, and algebras. The specific term abstract algebra
was coined at the beginning of the 20th century to distinguish this area from the other parts
of algebra. The term modern algebra has also been used to denote abstract algebra.
Two mathematical subject areas that study the properties of algebraic structures viewed
as a whole are universal algebra and category theory. Algebraic structures, together with
the associated homomorphisms, form categories. Category theory is a powerful formalism for
studying and comparing different algebraic structures.
History As in other parts of mathematics, concrete
problems and examples have played important roles in the development of abstract algebra.
Through the end of the nineteenth century, many -- perhaps most -- of these problems
were in some way related to the theory of algebraic equations. Major themes include:
Solving of systems of linear equations, which led to linear algebra
Attempts to find formulae for solutions of general polynomial equations of higher degree
that resulted in discovery of groups as abstract manifestations of symmetry
Arithmetical investigations of quadratic and higher degree forms and diophantine equations,
that directly produced the notions of a ring and ideal.
Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic
structures and then proceed to establish their properties. This creates a false impression
that in algebra axioms had come first and then served as a motivation and as a basis
of further study. The true order of historical development was almost exactly the opposite.
For example, the hypercomplex numbers of the nineteenth century had kinematic and physical
motivations but challenged comprehension. Most theories that are now recognized as parts
of 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. An archetypical example
of this progressive synthesis can be seen in the history of group theory.
Early group theory There were several threads in the early development
of group theory, in modern language loosely corresponding to number theory, theory of
equations, and geometry. Leonhard Euler considered algebraic operations
on numbers modulo an integer, modular arithmetic, in his generalization of Fermat's little theorem.
These investigations were taken much further by Carl Friedrich Gauss, who considered the
structure of multiplicative groups of residues mod n and established many properties of cyclic
and more general abelian groups that arise in this way. In his investigations of composition
of binary quadratic forms, Gauss explicitly stated the associative law for the composition
of forms, but like Euler before him, he seems to have been more interested in concrete results
than in general theory. In 1870, Leopold Kronecker gave a definition of an abelian group in the
context of ideal class groups of a number field, generalizing Gauss's work; but it appears
he did not tie his definition with previous work on groups, particularly permutation groups.
In 1882, considering the same question, Heinrich M. Weber realized the connection and gave
a similar definition that involved the cancellation property but omitted the existence of the
inverse element, which was sufficient in his context.
Permutations were studied by Joseph-Louis Lagrange in his 1770 paper Réflexions sur
la résolution algébrique des équations devoted to solutions of algebraic equations,
in which he introduced Lagrange resolvents. Lagrange's goal was to understand why equations
of third and fourth degree admit formulae for solutions, and he identified as key objects
permutations of the roots. An important novel step taken by Lagrange in this paper was the
abstract view of the roots, i.e. as symbols and not as numbers. However, he did not consider
composition of permutations. Serendipitously, the first edition of Edward Waring's Meditationes
Algebraicae appeared in the same year, with an expanded version published in 1782. Waring
proved the main theorem on symmetric functions, and specially considered the relation between
the roots of a quartic equation and its resolvent cubic. Mémoire sur la résolution des équations
of Alexandre Vandermonde developed the theory of symmetric functions from a slightly different
angle, but like Lagrange, with the goal of understanding solvability of algebraic equations.
Kronecker claimed in 1888 that the study of modern algebra began with this first paper
of Vandermonde. Cauchy states quite clearly that Vandermonde had priority over Lagrange
for this remarkable idea, which eventually led to the study of group theory.
Paolo Ruffini was the first person to develop the theory of permutation groups, and like
his predecessors, also in the context of solving algebraic equations. His goal was to establish
the impossibility of an algebraic solution to a general algebraic equation of degree
greater than four. En route to this goal he introduced the notion of the order of an element
of a group, conjugacy, the cycle decomposition of elements of permutation groups and the
notions of primitive and imprimitive and proved some important theorems relating these concepts,
such as if G is a subgroup of S5 whose order is divisible
by 5 then G contains an element of order 5. Note, however, that he got by without formalizing
the concept of a group, or even of a permutation group. The next step was taken by Évariste
Galois in 1832, although his work remained unpublished until 1846, when he considered
for the first time what is now called the closure property of a group of permutations,
which he expressed as ... if in such a group one has the substitutions
S and T then one has the substitution ST. The theory of permutation groups received
further far-reaching development in the hands of Augustin Cauchy and Camille Jordan, both
through introduction of new concepts and, primarily, a great wealth of results about
special classes of permutation groups and even some general theorems. Among other things,
Jordan defined a notion of isomorphism, still in the context of permutation groups and,
incidentally, it was he who put the term group in wide use.
The abstract notion of a group appeared for the first time in Arthur Cayley's papers in
1854. Cayley realized that a group need not be a permutation group, and may instead consist
of matrices, whose algebraic properties, such as multiplication and inverses, he systematically
investigated in succeeding years. Much later Cayley would revisit the question whether
abstract groups were more general than permutation groups, and establish that, in fact, any group
is isomorphic to a group of permutations. Modern algebra
The end of the 19th and the beginning of the 20th century saw a tremendous 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 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.
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 up 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 forever changed for the
mathematical world the meaning of the word algebra from the theory of equations to the
theory of algebraic structures. Basic concepts
By abstracting away various amounts of detail, mathematicians have created theories of various
algebraic structures that apply to many objects. 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; associativity, identity, and inverses; and other more complex structures.
With additional structure, more theorems could be proved, but the generality is reduced.
The "hierarchy" of algebraic objects creates a hierarchy of the corresponding theories:
for instance, the theorems of group theory apply to rings since a ring is a group over
one of its operations. Mathematicians choose a balance between the amount of generality
and the richness of the theory. Examples of algebraic structures with a single
binary operation are: Magmas
Quasigroups Monoids
Semigroups Groups
More complicated examples include: Rings
Fields Modules
Vector spaces Algebras over fields
Associative algebras Lie algebras
Lattices Boolean algebras
Applications 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 recently proved Poincaré conjecture 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.
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 dimension
of the Lie algebra, and these bosons interact with the force they mediate if the Lie algebra
is nonabelian. See also
Coding theory Publications in abstract algebra
References ^ Vandermonde biography in Mac Tutor History
of Mathematics Archive. ^ Schumm, Bruce, Deep Down Things, Baltimore:
Johns Hopkins University Press, ISBN 0-8018-7971-X Sources
Allenby, R.B.J.T., Rings, Fields and Groups, Butterworth-Heinemann, ISBN 978-0-340-54440-2
Artin, Michael, Algebra, Prentice Hall, ISBN 978-0-89871-510-1 Burris, Stanley N.; Sankappanavar, H. P. [1981],
A Course in Universal Algebra Gilbert, Jimmie; Gilbert, Linda, Elements
of Modern Algebra, Thomson Brooks/Cole, ISBN 978-0-534-40264-8 Lang, Serge, Algebra, Graduate Texts in Mathematics
211, New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
Sethuraman, B. A., 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., Guide to Abstract Algebra, Houndmills: Palgrave, ISBN 978-0-333-79447-0
W. Keith Nicholson Introduction to Abstract Algebra, 4th edition, John Wiley & Sons ISBN
978-1-118-13535-8 . John R. Durbin Modern Algebra : an introduction,
John Wiley & Sons External links
John Beachy: Abstract Algebra On Line, Comprehensive list of definitions and theorems.
Edwin Connell "Elements of Abstract and Linear Algebra ", Free online textbook.
Fredrick M. Goodman: Algebra: Abstract and Concrete.
Judson, Thomas W., Abstract Algebra: Theory and Applications An introductory undergraduate
text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral
domains, fields and Galois theory. Free downloadable PDF with open-source GFDL license.
Zeidler, A. Bernhard, Abstract Algebra A web-book on algebra and commutative algebra.
Warning: work in progress! Free downloadable PDF under Open Publication License.