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Abstract algebra
*** Shopping-Tip: Abstract algebra
see
Abstract algebra
:''This article is about the branch of mathematics. For other uses of the term "algebra" see
algebra (disambiguation).''
'''Abstract algebra''' is the field of
mathematics concerned with the study of
algebraic structures, such as
group (mathematics) groups,
ring (mathematics) rings,
field (mathematics) fields,
module (mathematics) modules,
vector spaces, and
algebra over a field algebras. Structures of this sort are defined formally, starting in the nineteenth century.
Abstract algebra, in its early life at the start of the twentieth century, was more often called ''modern algebra''. Its study was part of the drive for more
intellectual rigor in mathematics. Initially, the logical assumptions in classical
algebra, on which the whole of mathematics (and major parts of the
natural sciences) depend, were written out, as
axiomatic systems. On that basis disciplines such as
group theory and
ring theory took their places in
pure mathematics. The term ''abstract algebra'' is now used to distinguish the aggregate of such fields from the
elementary algebra ("high school algebra"), which teaches the correct rules for manipulating formulas and algebraic expressions involving
real numbers real and
complex numbers, and unknowns. Elementary algebra can be taken to be an introductory branch of
commutative algebra.
Contemporary mathematics and
mathematical physics constantly and intensively use the results of abstract algebra; for example, the theory of
Lie algebras, an abstract structure only isolated towards the end of the nineteenth century by
Sophus Lie. Fields such as
algebraic number theory,
algebraic topology and
algebraic geometry apply algebraic methods in other areas. The idea of
representation theory in mathematics is, roughly speaking,to take the 'abstract' out of 'abstract algebra', studying the concrete side of a given structure.
The term '''abstract algebra''' is sometimes used in
universal algebra, a general theory of algebra, where most authors use simply the term "algebra".
History and examples
Historically, algebraic structures usually arose first in some other field of mathematics, were specified axiomatically, and were then studied in their own right in abstract algebra. Because of this, abstract algebra has numerous fruitful connections to all other branches of mathematics.
Examples of algebraic structures with a single
binary operation are:
*
magma (algebra) magmas,
*
quasigroups,
*
monoids,
semigroups and, most important,
group (mathematics) groups.
More complicated examples include:
*
ring (mathematics) rings and
field (mathematics) fields
*
module (mathematics) modules and
vector spaces
*
algebra over a field algebras over fields
*
associative algebras and
Lie algebras
*
lattice (order) lattices and
Boolean algebras
In
universal algebra, all those definitions and facts are collected that apply to all algebraic structures alike. All the above classes of objects, together with the proper notion of
homomorphism, form
category theory categories, and category theory frequently provides the formalism for translating between and comparing different algebraic structures.
An example
The systematic study of algebra has allowed mathematicians to bring under a common logical description apparently disparate conceptions. For example, consider two rather distinct operations: the composition of
function composition functions, f(g(x)), and the multiplication of
matrix multiplication matrices, AB. These two operations are, in fact, the same. To see this, think about multiplying two square matrices (AB) by a one-column vector, x. This, in fact, defines a function that is equivalent to composing Ay with Bx: Ay = A(Bx) = (AB)x. Functions under composition and matrices under multiplication form sets called
monoids; a monoid under an operation is associative for all its elements ( (ab)c = a(bc) ) and contains an element e such that, for any a, ae = ea = a.
See also
*
List of publications in mathematics#Abstract algebra Important publications in abstract algebra
Further reading
* {{cite book | author=Sethuraman, B. A. | title=Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility | publisher=Springer | year=1996 | id=ISBN 0-387-94848-1}}
External links
{{book}}
* John Beachy: ''[http://www.math.niu.edu/~beachy/aaol/contents.html Abstract Algebra On Line]'', Comprehensive list of definitions and theorems.
* Joseph Mileti: ''Mathematics Museum: [http://www.math.uchicago.edu/~mileti/museum/algebra.html Abstract Algebra]'', A good introduction to the subject in real-life terms.
{{Mathematics-footer}}
Category:Abstract algebra
ar:جبر تجريدي
da:Abstrakt algebra
de:Abstrakte Algebra
es:Ã?lgebra abstracta
fa:جبر مجرد
fr:Algèbre générale
ko:추�대수학
io:Abstrakta algebro
it:Algebra astratta
he:×?לגברה מופשטת
nl:Abstracte algebra
no:Abstrakt algebra
nn:Abstrakt algebra
pt:Ã?lgebra abstrata
ru:Ð?бÑ?трактнаÑ? алгебра
fi:Abstrakti algebra
sv:Abstrakt algebra
th:พีชคณิตนามธรรม
vi:Ä?ại số trừu tượng
zh:抽象代数
{{commonscat|Abstract algebra}}
'''
Abstract algebra''' is the field of
mathematics concerned with the study of
algebraic structures such as
group (mathematics) groups,
ring (mathematics) rings and
field (mathematics) fields. The term "abstract algebra" is used to distinguish the field from "
elementary algebra" or "high school algebra" which teaches the correct rules for manipulating formulas and algebraic expressions involving
real numbers real and
complex numbers.
Related categories
*
:Category:Algebraic geometry Algebraic geometry
Category:Algebra
bg:КатегориÑ?:Ð?бÑ?трактна алгебра
es:CategorÃa:Ã?lgebra abstracta
fr:Catégorie:Algèbre générale
ko:분류:추�대수학
it:Categoria:Algebra astratta
ru:КатегориÑ?:Ð?бÑ?трактнаÑ? алгебра
sl:Category:Abstraktna algebra
sv:Kategori:Abstrakt algebra
vi:Category:Ä?ại số trừu tượng
zh:Category:抽象代数
uk:КатегоріÑ?:Ð?бÑ?трактна алгебра
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