Abstract algebra

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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

Further reading

  • Sethuraman, B. A. (1996). Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility, Springer. ISBN 0-387-94848-1.

External links

Wikibooks has more about this subject:

de:Abstrakte Algebra es:Álgebra abstracta fr:Algèbre abstraite ko:추상대수학 it:Algebra astratta he:אלגברה מופשטת nl:Abstracte algebra no:Abstrakt algebra nn:Abstrakt algebra pt:Álgebra abstrata ru:Абстрактная алгебра fi:Abstrakti algebra sv:Abstrakt algebra th:พีชคณิตนามธรรม vi:Đại số trừu tượng zh:抽象代数