# Algebra

Algebra is one of the fundamental subsections of mathematics.

## word history

one of the first representations of algebra is the Aryabhattiya, a mathematical text book of the Indian mathematician Aryabhatta from that 5. Century; used methodology was called Bijaganitam. In 13. Century took over and refined the Arabs this method and called it aluminium-jabr (“joining more broken (bone) parts”), which from the title of the computing text book Hisab aluminium-dschabr wa-l-muqabala the Persian mathematician aluminium-Khwarizmiis taken. Four centuries after the appearance of the book appeared its latin translation Ludus algebrae et almucgrabalaeque. Out aluminium-jabr the word “algebra” shortened today developed.

## algebra as subsection of mathematics: Definition and arrangement

contentsand methods of algebra have themselves in the course of history so strongly extended that it became heavy to indicate in a scarce definition what algebra actually are. Also it would be not practicably, all aspects of algebra in an encyclopedia article tootreat. We do not differentiate between therefore the following, by any means sharply from each other defined subsections:

• Elementary algebra is algebra in the sense of school mathematics. It covers the arithmetic rules of the natural, whole, broken and real numbers, handling expressions, which contain variables, and waysto the solution of simple algebraic equations.
• Classical algebra concerns itself with the release of general algebraic equations (see below) over the real or complex numbers. Their central result is the fundamental principle of the algebra, which it means that each not-constant polynomial n - ten degree into n linear factors with complex coefficients to be divided knows.
• Linear algebra treats the release of linear sets of equations, the investigation of vector spaces and the regulation of eigenvalues; it is basis for analytic geometry.
• Those multi-linear algebra acts of tensors.
• Abstract algebra is a basic discipline of modern mathematics. It concerns itself with algebraic structures such as groups, rings, bodies and their linkage.
• Computer algebra concerns itself with the symbolic manipulation of algebraic expressions.Accurate counting on whole, rational and algebraic numbers as well as on polynomials over these ranges of numbers forms an emphasis. On the theoretical side the search for efficient algorithms as well as the determination of the complexity of these algorithms are to be assigned to this subsection. On thatpractical side a multiplicity of computer algebra systems was developed, which make the computer-aided manipulation possible of algebraic expressions.
• Real algebra examines algebraic number bodies, on which an arrangement can be defined. For it positive polynomials are continued to examine.

## algebra as mathematicalStructure

as algebra (also: Algebraic structure) one designates also the Grundkonstrukt of abstract algebra: a quantity, on which one or more linkages are defined and apply into certain axioms. Groups, rings, bodies are thus examples of specialAlgebras.

To “algebra” designation also concrete algebraic structures, which are Verallgemeinerungen of the ring term, see algebra (structure).

## “algebraic” as attribute of numbers, functions, equations

algebraic as mathematical attribute has the following meanings:

• An algebraic equation is only finite an equation, to their formulation many elementary arithmetic operations (addition,Subtraction, multiplication, division) are necessary, in which thus for example no typical analytic functions occur.
• One receives the algebraic numbers as zeros from polynomials with rational coefficients; the quantity of the algebraic numbers forms the algebraic conclusion of the quantity of the rationalNumbers.
• The algebraic element extends the term of the algebraic number on zeros of polynomials with coefficients from any given body…

## literature

it gives many good text books to algebra. Exemplarily are here mentioned:

• Hans Kreul: Mathematics easily made, Harri German publishing house 2002. (Bases of algebra/no substantial previous knowledge necessarily, very understandably) ISBN 3-8171-1678-0
• S. Long: Algebra. Revised 3rd edition, Springer publishing house 2002.ISBN 038795385X
extensive standard work with many resuming notes and tasks. The representation is possibly too abstract for a first entrance.
• B. L. van the Waerden: Algebra I, II. Berlin, Springer publishing house 1993. ISBN 3-540-56801-8
the classical author, for its first expenditures into the 1930er years still the title modern trend algebra carried and that for the first time consistently the axiomatic beginning of E. Noether represented. In the language in the meantime somewhat becomes outdated.

 Wiktionary: Algebra - word origin, synonyms and translations