### ALGEBRA RESEARCH

After the crit on the 9th of December, with my peers and third year students, I decided to look briefly into algebra to see how this can be translated into a typographical concept relative to parentheses.

## algebra |ˈaljəbrə|nounthe part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.• a system of this based on given axioms.

## How to distinguish between different meanings of "algebra"

For historical reasons, the word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. Such a situation, where a single word has many meanings in the same area of mathematics, may be confusing. However the distinction is easier if one recalls that the name of a scientific area is usually singular and without an article and the name of a specific structure requires an article or the plural. Thus we have:

- As a single word without article, "algebra" names a broad part of mathematics (see below).
- As a single word with article or in plural, "algebra" denotes a specific mathematical structure. See algebra (ring theory) and algebra over a field.
- With a qualifier, there is the same distinction:
- Without article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part ofprimary and secondary education), or abstract algebra (the study of the algebraic structures for themselves).
- With an article, it means an instance of some abstract structure, like a Lie algebra or an associative algebra.
- Frequently both meanings exist for the same qualifier, like in the sentence:
*Commutative algebra is the study of commutative rings, that all are commutative algebras over the integers*.

- Sometimes "algebra" is also used to denote the operations and methods related to algebra in the study of a structure that does not belong to algebra. For example
*algebra ofinfinite series*may denote the methods for computing with series without using the notions of infinite summation, limits and convergence.

## Algebra as a branch of mathematics

Algebra can essentially be considered as doing computations similar to that of arithmetic with non-numerical mathematical objects.

^{[1]}Initially, these objects represented either numbers that were not yet known (*unknowns*) or unspecified numbers (*indeterminates*or*parameters*), allowing one to state and prove properties that are true no matter which numbers are involved. For example, in the quadratic equation
are indeterminates and is the unknown. Solving this equation amounts to computing with the variables to express the unknown in terms of the indeterminates. Then, substituting any numbers for the indeterminates, gives the solution of a particular equation after a simple arithmetic computation.

As it developed, algebra was extended to other non-numerical objects, like vectors, matrices or polynomials. Then, the structural properties of these non-numerical objects were abstracted to define algebraic structures like groups, rings, fields and algebras.

Before the 16th century, mathematics was divided into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from 16th or 17th century. From the second half of 19th century on, many new fields of mathematics appeared, some of them included in algebra, either totally or partially.

It follows that algebra, instead of being a true branch of mathematics, appears nowadays, to be a collection of branches sharing common methods. This is clearly seen in theMathematics Subject Classification

^{[2]}where none of the first level areas (two digit entries) is called*algebra*. In fact, algebra is, roughly speaking, the union of sections 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Some other first level areas may be considered to belong partially to algebra, like 11-Number theory (mainly for algebraic number theory) and 14-Algebraic geometry.
Elementary algebra is the part of algebra that is usually taught in elementary courses of mathematics.

Abstract algebra is a name usually given to the study of the algebraic structures themselves.

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