Abstract algebra
This article is about
the branch of mathematics. For the Swedish band, see Abstract Algebra.
"Modern
algebra" redirects here. For van der Waerden's book, see Moderne Algebra.
In algebra, which is a broad
division of mathematics, abstract
algebra (occasionally called modern algebra) is the study
of algebraic structures.
Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras.
The term abstract algebra was coined in the early 20th century
to distinguish this area of study from the other parts of algebra.
Algebraic structures, with their
associated homomorphisms,
form mathematical
categories. Category theory is a
formalism that allows a unified way for expressing properties and constructions
that are similar for various structures.
Universal algebra is a
related subject that studies types of algebraic structures as single objects.
For example, the structure of groups is a single object in universal algebra,
which is called variety of groups.
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 (finite group)
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 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.
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.
Examples involving several operations include:
·
Ring
·
Field
·
Module
·
Vector space
·
Algebra over a field
·
Associative algebra
·
Lie algebra
·
Lattice
·
Boolean algebra
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 Poincaré 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.
·
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.
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