Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with Most contemporary authors simply write algebra instead of abstract algebra.

The term abstract algebra now refers to the study of all algebraic structures, as distinct from the elementary algebra ordinarily taught to children, which teaches the correct rules for manipulating formulas and algebraic expressions involving real and complex numbers, and unknowns. Elementary algebra is a fundamental and relatively basic form of Algebra taught to students who are presumed to have little or no formal knowledge of Mathematics beyond In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Elementary algebra can be taken as an informal introduction to the structures known as the real field and commutative algebra. Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings

Contemporary mathematics and mathematical physics make intensive use of abstract algebra; for example, theoretical physics draws on Lie algebras. Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Subject areas such as algebraic number theory, algebraic topology, and algebraic geometry apply algebraic methods to other areas of mathematics. In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Representation theory, roughly speaking, takes the 'abstract' out of 'abstract algebra', studying the concrete side of a given structure; see model theory. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models

Two mathematical subject areas that study the properties of algebraic structures viewed as a whole are universal algebra and category theory. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Algebraic structures, together with the associated homomorphisms, form categories. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships Category theory is a powerful formalism for studying and comparing different algebraic structures.

History and examples

As in other parts of mathematics, concrete problems and examples have played important roles in the development of algebra. Through the end of the nineteenth century many, perhaps most, of these problems were in some way related to the theory of algebraic equations. Among major themes we can mention:

• solving of systems of linear equations, which led to matrices, determinants and linear algebra. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n Linear algebra is the branch of Mathematics concerned with
• attempts to find formulas for solutions of general polynomial equations of higher degree that resulted in discovery of groups as abstract manifestations of symmetry;
• and arithmetical investigations of quadratic and higher degree forms and diophantine equations, notably, in proving Fermat's last theorem, that directly produced the notions of a ring and ideal. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.

Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then proceed to establish their properties, creating a false impression that somehow in algebra axioms had come first and then served as a motivation and as a basis of further study. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, The true order of historical development was almost exactly the opposite. Most theories that we now recognize as parts of algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. An archetypical example of this progressive synthesis can be seen in the theory of groups. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups.

Early group theory

There were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, and geometry, of which we concentrate on the first two.

Leonhard Euler considered algebraic operations on numbers modulo an integer, modular arithmetic, proving his generalization of Fermat's little theorem. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers In Number theory, Euler's theorem (also known as the Fermat-Euler theorem or Euler's totient theorem) states that if n is a positive Integer Fermat's little theorem (not to be confused with Fermat's last theorem) states that if p is a Prime number, then for any Integer a These investigations were taken much further by Carl Friedrich Gauss, who considered the structure of multiplicative groups of residues mod n and established many properties of cyclic and more general abelian groups that arise in this way. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In his investigations of composition of binary quadratic forms, Gauss explicitly stated the associative law for the composition of forms, but like Euler before him, he seems to have been more interested in concrete results than in general theory. In Mathematics, associativity is a property that a Binary operation can have In 1870, Leopold Kronecker gave a definition of an abelian group in the context of ideal class groups of a number field, a far-reaching generalization of Gauss's work. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued In Mathematics, the extent to which Unique factorization fails in the ring of integers of an Algebraic number field (or more generally any Dedekind domain It appears that he did not tie it with previous work on groups, in particular, permutation groups. In 1882 considering the same question, Heinrich M. Weber realized the connection and gave a similar definition that involved the cancellation property but omitted the existence of the inverse element, which was sufficient in his context (finite groups). Heinrich Martin Weber ( 5 March 1842 &ndash 17 May 1913) was a German Mathematician who specialized in Algebra In Mathematics, the notion of cancellative is a generalization of the notion of Invertible. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to

Permutations were studied by Joseph Lagrange in his 1770 paper Réflexions sur la résolution algébrique des équations devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. Lagrange's goal was to understand why equations of third and fourth degree admit formulas for solutions, and he identified as key objects permutations of the roots. An important novel step taken by Lagrange in this paper was the abstract view of the roots, i. e. as symbols and not as numbers. However, he did not consider composition of permutations. Serendipitously, the first edition of Edward Waring's Meditationes Algebraicae appeared in the same year, with an expanded version published in 1782. Edward Waring (1736 – August 15, 1798) was an English Mathematician who was born in Old Heath (near Shrewsbury) Waring proved the main theorem on symmetric functions, and specially considered the relation between the roots of a quartic equation and its resolvent cubic. In Mathematics, specifically in Commutative algebra, the elementary symmetric polynomials are one type of basic building block for Symmetric polynomials Mémoire sur la résolution des équations of Alexandre Vandermonde (1771) developed the theory of symmetric functions from a slightly different angle, but like Lagrange, with the goal of understanding solvability of algebraic equations. Alexandre-Théophile Vandermonde ( 28 February 1735 – 1 January 1796) was a French Musician and Chemist who

Kronecker claimed in 1888 that the study of modern algebra began with this first paper of Vandermonde. Cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea which eventually led to the study of group theory. ^[1]

Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations. Paolo Ruffini ( September 22, 1765 – May 9, 1822) was an Italian Mathematician and Philosopher. In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation His goal was to establish impossibility of algebraic solution to a general algebraic equation of degree greater than four. En route to this goal he introduced the notion of the order of an element of a group, conjugacy, the cycle decomposition of elements of permutation groups and the notions of primitive and imprimitive and proved some important theorems relating these concepts, such as

if G is a subgroup of S₅ whose order is divisible by 5 then G contains an element of order 5.

Note, however, that he got by without formalizing the concept of a group, or even of a permutation group. The next step was taken by Évariste Galois in 1832, although his work remained unpublished until 1846, when he considered for the first time what we now call the closure property of a group of permutations, which he expressed as

. . . if in such a group one has the substitutions S and T then one has the substitution ST.

The theory of permutation groups received further far-reaching development in the hands of Augustin Cauchy and Camille Jordan, both through introduction of new concepts and, primarily, a great wealth of results about special classes of permutation groups and even some general theorems. Marie Ennemond Camille Jordan ( January 5 1838 &ndash January 22 1922) was a French Mathematician, known both for his foundational Among other things, Jordan defined a notion of isomorphism, still in the context of permutation groups and, incidentally, it was he who put the term group in wide use. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

The abstract notion of a group appeared for the first time in Arthur Cayley's papers in 1854. Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician. Cayley realized that a group need not be a permutation group (or even finite), and may instead consist of matrices, whose algebraic properties, such as multiplication and inverses, he systematically investigated in succeeding years. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Much later Cayley would revisit the question whether abstract groups were more general than permutation groups, and establish that, in fact, any group is isomorphic to a group of permutations.

Modern algebra

The end of 19th and the beginning of the 20th century saw a tremendous shift in methodology of mathematics. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of an abstract group. Questions of structure and classification of various mathematical objects came to forefront. These processes were occurring throughout all of mathematics, but became especially pronounced in algebra. Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such as groups, rings, and fields. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division The algebraic investigations of general fields by Ernst Steinitz and of commutative and then general rings by David Hilbert, Emil Artin and Emmy Noether, building up on the work of Ernst Kummer, Leopold Kronecker and Richard Dedekind, who had considered ideals in commutative rings, and of Georg Frobenius and Issai Schur, concerning representation theory of groups, came to define abstract algebra. Ernst Steinitz ( June 13, 1871 – September 29 1928) was a German Mathematician. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Emil Artin ( March 3, 1898, in Vienna – December 20, 1962, in Hamburg) was an Austrian Mathematician Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Ferdinand Georg Frobenius ( October 26, 1849 – August 3, 1917) was a German Mathematician, best-known for his contributions Issai Schur ( January 10, 1875 in Mogilyov &ndash January 10, 1941 in Tel Aviv) was a Mathematician who These developments of the last quarter of the 19th century and the first quarter of 20th century were systematically exposed in Bartel van der Waerden's Moderne algebra, the two-volume monograph published in 1930–1931 that forever changed for the mathematical world the meaning of the word algebra from the theory of equations to the theory of algebraic structures. Bartel Leendert van der Waerden ( February 2 1903, Amsterdam, Netherlands – January 12 1996, Zürich,

An example

Abstract algebra facilitates the study of properties and patterns that seemingly disparate mathematical concepts have in common. For example, consider the distinct operations of function composition, f(g(x)), and of matrix multiplication, AB. In Mathematics, a composite function represents the application of one function to the results of another In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix These two operations have, in fact, the same structure. To see this, think about multiplying two square matrices, AB, by a one column vector, x. This defines a function equivalent to composing Ay with Bx: Ay = A(Bx) = (AB)x. Functions under composition and matrices under multiplication are examples of monoids. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation A set S and a binary operation over S, denoted by concatenation, form a monoid if the operation associates, (ab)c = a(bc), and if there exists an e∈S, such that ae = ea = a. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In Mathematics, associativity is a property that a Binary operation can have

See also

• Universal algebra
• Coding theory
• Important publications in abstract algebra

References and further reading

1. ^ Vandermonde biography in Mac Tutor History of Mathematics Archive. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" Coding theory is one of the most important and direct applications of Information theory. Algebra Theory of equations Hisab

• 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.  
• Jimmie Gilbert, Linda Gilbert (2005). Elements of Modern Algebra. Thomson Brooks/Cole. ISBN 0-534-40264-X.  
• R. B. J. T. Allenby (1991). Rings, Fields and Groups. Butterworth-Heinemann. ISBN 0-340-54440-6.  
• C. Whitehead (2002). Guide2 Abstract Algebra (2nd edition). ISBN 0-333-79447-8;.  

A monograph available free online:

• Burris, Stanley N. & Sankappanavar, H. P. (1999), A Course in Universal Algebra, <http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html> 

External links

• Fredrick M. Goodman: Algebra: Abstract and Concrete.
• John Beachy: Abstract Algebra On Line, Comprehensive list of definitions and theorems.
• Joseph Mileti: Mathematics Museum: Abstract Algebra, A good introduction to the subject in real-life terms.
• Edwin Connell "[1]", Free online textbook.