Algebra(2)

The branch of mathematics in which one studies algebraic operations (cf. Algebraic operation).

Historical survey.

The simplest algebraic operations — arithmetic operations on positive integers and positive rational numbers — can be encountered in the oldest mathematical texts, which indicates that the principal properties of these operations were known even in early antiquity. In particular, the Arithmetic of Diophantus (3rd century A.D.) had a major influence on the development of algebraic ideas and symbols. The term "Algebra" originates from the work Al jabr al-muqabala by Mohammed Al-Khwarizmi (9th century A.D.), which describes general methods for solving problems which can be reduced to algebraic equations of the first and second degree. Towards the end of the 15th century the cumbersome verbal descriptions of mathematical operations which had previously prevailed began to be replaced by the contemporary symbols "+" and "-" , and subsequently symbols for powers, roots, and parentheses appeared. F. Viète, at the end of the 16th century, was the first to use the letters of the alphabet to denote the constants and the variables in a problem. Most of the present-day symbols of algebra were known as early as the mid-17th century, which marks the end of the "prehistory" of algebra. The development of algebra proper took place during the next three centuries, during which views as to the proper subject matter of this discipline kept radically changing.

In the 17th century and 18th century "Algebra" was understood to mean the science of computations carried out on algebraic symbols — "identity" transformations of formulas consisting of letters, solving algebraic equations (cf. Algebraic equation), etc. — as distinct from arithmetic, which dealt with calculations performed on explicit numbers. It was assumed, however, that the symbols stood for actual numbers: integers or fractions. A brief table of the contents of one of the best textbooks of that time, L. Euler's Introduction to algebra includes integers, ordinary and decimal fractions, roots, logarithms, algebraic equations of degrees one to four, progressions, additions, Newton's binomial and Diophantine equations. Thus, by the mid-18th century, algebra corresponded, more or less, to the "elementary" algebra of our own days.

The principal subject dealt with by the algebra of the 18th century and 19th century were polynomials. Historically, the first problem was the solution of algebraic equations in one unknown, i.e. equations of the type:

$$ a _ {0} x ^ {n} + a _ {1} x ^ {n-1 } + \dots + a _ {n} = 0 . $$

The purpose was to derive formulas expressing the roots of the equation in terms of its coefficients, by means of addition, multiplication, subtraction, division and extraction of roots ( "solution by radicalssolution by radicals" ). Mathematicians were able to solve first- and second-degree equations even in the earliest times. Substantial advances were made in the 16th century by Italian mathematicians: a formula was found for solving third-degree equations (cf. Cardano formula) and fourth-degree equations (cf. Ferrari method). During the following three centuries fruitless efforts were made to find similar formulas for solving equations of higher degrees; in this connection, the problem of finding at least a "formula-free" proof of the existence of a complex root of an arbitrary algebraic equation with complex coefficients became of major interest. This theorem was first stated in the 17th century by A. Girard, but was rigorously proved by C.F. Gauss only towards the end of the 18th century (cf. Algebra, fundamental theorem of). Finally, it was established by N.H. Abel in 1824 that equations of degree higher than four cannot, in general, be solved by radicals, and E. Galois in 1830 stated a general criterion for the solvability of algebraic equations by radicals (cf. Galois theory). Other problems were neglected at that time, and algebra was understood to mean the "analysis of equations" , as noted by J. Serret in his course of higher algebra (1849).

Studies on algebraic equations in one unknown were accompanied by studies on algebraic equations in several unknowns, in particular of systems of linear equations. The study of linear equations resulted in the introduction of the concepts of a matrix and a determinant. Matrices subsequently became the subject of an independent theory, the algebra of matrices, and their scope of application was extended beyond the solution of systems of linear equations.

From the mid-19th century onwards studies in algebra gradually moved away from the theory of equations towards the study of arbitrary algebraic operations. The first attempts at an axiomatic study of algebraic operations dates back to the "theory of relations" of Euclid, but no progress was made in this direction, since a geometrical interpretation of even the simplest arithmetic operations — ratios of lengths or of areas — is impossible. Further progress only became possible as a result of gradual generalization and intensive study of the concept of a number, and of the appearance of arithmetic operations performed on objects entirely unlike any number. The first such examples were Gauss' "composition of binary quadratic forms" , and P. Ruffini's and A.L. Cauchy's multiplication of permutations. The abstract concept of an algebraic operation appeared in the mid-19th century in the context of studies on complex numbers (cf. Complex number). There appeared G. Boole's algebra of logic, H. Grassmann's exterior algebra, W. Hamilton's quaternions (cf. Quaternion) and A. Cayley's matrix calculus, while C. Jordan published a major treatise on permutation groups.

These studies prepared the way for the transition of algebra at the turn of the 19th century into its modern stage of development, which is characterized by the combination of previously separate algebraic ideas on a common axiomatic basis and by a considerable extension of the scope of its applications. The modern view of algebra, and of the general theory of algebraic operations, crystallized at the beginning of the 20th century under the influence of D. Hilbert, E. Steinitz, E. Artin and E. Noether, and was fully established by 1930 with the appearance of B.L. van der Waerden's Modern algebra.

The subject matter of algebra, its principal branches and its connection with other branches of mathematics.

The subject matter of modern algebra are sets and algebraic operations on these sets (i.e. algebras or universal algebras, cf. the terminology in Algebra; Universal algebra), considered up to an isomorphism. This means that, from the point of view of algebra, the sets themselves and the sets as carriers of algebraic operations are indistinguishable, and in this sense the proper subject of study are the algebraic operations themselves.

For a long time the studies that were actually carried out concerned only a few basic types of universal algebras which naturally appeared in the development of mathematics and its applications.

One of the most important and most thoroughly studied type of algebras is a group, i.e. an algebra with one associative binary operation, containing a unit element and, for each element, an inverse element. The concept of a group was, historically, the first example of a universal algebra and in fact served, in many respects, as a model for the construction of algebra and of mathematics in general at the turn of the 19th century. Independent studies on generalizations of groups such as semi-groups, quasi-groups and loops began only much later (cf. Loop; Quasi-group; Semi-group).

Rings and fields are very important types of algebras with two binary operations. The operations in rings and fields are usually called addition and multiplication. A ring is defined by Abelian group axioms for the addition, and by distributive laws for multiplication with respect to addition (cf. Rings and algebras). Originally, only rings with associativity multiplication were studied, and this requirement of associativeness occasionally forms part of the definition of a ring (cf. Associative rings and algebras). The study of non-associative rings (cf. Non-associative rings and algebras) is today a fully recognized independent discipline. A skew-field is an associative ring in which the set of all non-zero elements is a multiplicative group. A field is a skew-field in which multiplication is commutative. Number fields, i.e. sets of numbers closed under addition, multiplication, subtraction and division by non-zero numbers, were implicitly included in the very first studies of algebraic equations. Associative-commutative rings and fields are the main objects studied in commutative algebra and in the closely related field algebraic geometry.

Another important type of algebra with two binary operations is a lattice. Typical examples of lattices include: the system of subsets of a given set with the operations of set-theoretic union and intersection, and the set of positive integers with the operations of taking the least common multiple and the greatest common divisor.

Linear (or vector) spaces over a field may be treated as universal algebras with one binary operation (addition) and with a collection of unary operations (multiplication by scalars of the ground field). Linear spaces over skew-fields have also been studied. If a ring is considered instead of a set of scalars, the more general concept of a module is obtained. An important part of algebra, linear algebra, studies linear spaces, modules and their linear transformations, as well as problems related to them. A part of it, the theory of linear equations and the theory of matrices, was formulated as early as the 19th century. A closely related subject is that of multilinear algebra.

Initial studies on the general theory of arbitrary universal algebras (this theory is sometimes called "universal algebra" ) date back to the 1930s and were carried out by G. Birkhoff. At the same time A.I. Mal'tsev and A. Tarski laid the foundations for the theory of models (cf. Model (in logic)), i.e. sets with marked relations on them. Subsequently, the theory of universal algebras and the theory of models became so closely linked that they gave rise to a new discipline, intermediate between algebra and mathematical logic, called the theory of algebraic systems (cf. Algebraic system), the subject of which are sets with algebraic operations and relations defined on them.

A number of disciplines intermediate between algebra and other fields of mathematics have been created by the introduction into universal algebras of complementary structures compatible with the algebraic operations. These include topological algebra (including the theory of topological groups and Lie groups, cf. Lie group; Topological group), the theory of normed rings (cf. Normed ring), differential algebra, and theories of various ordered algebraic formations. Homological algebra, which originates both from algebra and from topology, arose in the 1950s as a discipline in its own right.

The role of algebra in modern mathematics is extremely important, and there is a trend towards further "algebraization" of mathematics. A typical way of studying many mathematical objects that are sometimes far removed from algebra is to construct algebraic systems which adequately represent the behaviour of these objects. Thus, the study of Lie groups can be largely reduced to the study of their algebraic counterparts: Lie algebras (cf. Lie algebra). A similar method is used in topology: to each topological space is assigned, in some standard manner, an infinite series of homology groups (cf. Homology group), and these series of algebraic "reflections of them" make it possible to evaluate, very accurately, the properties of the spaces themselves. The recent major discoveries in topology were made using algebra as a tool (cf. Algebraic topology).

It would appear at first sight that the translation of problems into the language of algebra, solving them in this language and translating them back is merely a superfluous complication. In fact, such a method turns out to be highly convenient, and occasionally the only possible one. This is because by algebraization one solves problems not only by purely verbal considerations, but also by using the powerful apparatus of formal algebraic calculations, so that one may occasionally overcome highly involved complications. This role of algebra in mathematics may be compared with the role of modern computers in the solution of practical problems.

Algebraic concepts and methods are widely employed in number theory (cf. Algebraic number theory), in functional analysis, in the theory of differential equations, in geometry (cf. Invariants, theory of; Projective geometry; Tensor algebra), and in other mathematical disciplines.

Besides its fundamental role in mathematics, algebra is very important from the point of view of its applications; examples are applications within physics (the representation theory of finite groups in quantum mechanics; discrete groups in crystallography), cybernetics (cf. Automata, theory of) and mathematical economics (linear inequalities, cf. Linear inequality).

For references, see also the articles on individual algebraic disciplines.

References

[1]	, The history of mathematics from Antiquity to the beginning of the XIX-th century , 1–3 , Moscow (1970–1972) (In Russian)
[2]	A.I. Mal'tsev, "On the history of algebra in the USSR during her first twenty-five years" Algebra and Logic , 10 : 1 (1971) pp. 68–75 Algebra i Logika , 10 : 1 (1971) pp. 103–118 Zbl 0234.01004
[3]	, Mathematics, its content, methods and meaning , 1–3 , Moscow (1956) (In Russian) Zbl 1049.00004 Zbl 0232.00001 Zbl 0111.00103
[4]	A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) MR0945393 MR0926059 MR0778202 MR0759341 MR0628003 MR0384363 Zbl 0237.13001
[5]	N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207
[6]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) MR1541390 Zbl 1032.00002 Zbl 1032.00001 Zbl 0903.01009 Zbl 0781.12003 Zbl 0781.12002 Zbl 0724.12002 Zbl 0724.12001 Zbl 0569.01001 Zbl 0534.01001 Zbl 0997.00502 Zbl 0997.00501 Zbl 0316.22001 Zbl 0297.01014 Zbl 0221.12001 Zbl 0192.33002 Zbl 0137.25403 Zbl 0136.24505 Zbl 0087.25903 Zbl 0192.33001 Zbl 0067.00502
[7]	S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[8]	A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian) Zbl 0266.08001

References

[a1]	R. Lidl, G. Pilz, "Applied abstract algebra" , Springer (1984) MR0765220 Zbl 0572.00001
[a2]	N. Jacobson, "Lectures in abstract algebra" , 1–3 , v. Nostrand (1951–1964) MR0392906 MR0392227 MR0369381 MR0172871 MR0053905 MR1570588 MR0041102 Zbl 0455.12001 Zbl 0326.00001 Zbl 0322.12001 Zbl 0314.15001 Zbl 0124.27002 Zbl 0053.21204