Information AboutAbstract Algebra |
| CATEGORIES ABOUT ABSTRACT ALGEBRA | |
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Abstract algebra is the field of Mathematics concerned with the study of Algebraic Structure s, such as Groups , Rings , Fields , Modules , Vector Space s, 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 System s. 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 Number s, 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 Algebra s, 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:
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. 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 s; 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 A monograph available free online:
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