Abstract algebra
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:
- magmas,
- quasigroups,
- monoids, semigroups and, most important, groups.
More complicated examples include:
- rings and fields
- modules and vector spaces
- 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.
See also
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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