Abstract algebra
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- 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 groups, rings, fields, modules, vector spaces, and 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 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".
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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:
- magmas,
- quasigroups,
- monoids, semigroups and, most important, groups.
More complicated examples include:
- rings and fields
- modules and vector spaces
- algebras over fields
- associative algebras and Lie algebras
- 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 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 functions, f(g(x)), and the multiplication of 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
References and further reading
External links
- John Beachy: Abstract Algebra On Line, Comprehensive list of definitions and theorems.
- Joseph Mileti: Mathematics Museum: Abstract Algebra, A good introduction to the subject in real-life terms.
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