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Abstract algebra

From Academic Kids

Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. The term abstract algebra is used to distinguish the field from "elementary algebra" or "high school algebra", which teach the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers, and unknowns. Abstract algebra was at times in the first half of the twentieth century known as modern algebra.

The term abstract algebra is sometimes used in universal 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:

More complicated examples include:

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.

See also

External links

Template:Book

  • John Beachy: Abstract Algebra On Line (http://www.math.niu.edu/~beachy/aaol/contents.html), Comprehensive list of definitions and theorems.
  • Joseph Mileti: Mathematics Museum: Abstract Algebra (http://www.math.uchicago.edu/~mileti/museum/algebra.html), A good introduction to the subject in real-life terms.

Topics in mathematics related to structure

Edit (http://en.wikipedia.org/w/wiki.phtml?title=Template:Structure&action=edit)
Abstract algebra | Universal algebra | Graph theory | Category theory | Order theory | Model theory | Structural proof theory
Geometry | Topology | General topology | Algebraic geometry | Algebraic topology | Differential geometry and topology
Analysis | Measure theory | Functional analysis | Harmonic analysis
de:Abstrakte Algebra

es:lgebra abstracta fr:Algbre abstraite ko:추상대수학 it:Algebra astratta he:אלגברה מופשטת nl:Abstracte algebra no:Abstrakt algebra nn:Abstrakt algebra ru:Абстрактная алгебра fi:Abstrakti algebra th:พีชคณิตนามธรรม zh:抽象代数

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