Abstract algebra

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This article is about the branch of mathematics. Abstract algebra_sentence_0

For the Swedish band, see Abstrakt Algebra. Abstract algebra_sentence_1

"Modern algebra" redirects here. Abstract algebra_sentence_2

For van der Waerden's book, see Moderne Algebra. Abstract algebra_sentence_3

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Abstract algebra_sentence_4

Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. Abstract algebra_sentence_5

The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra. Abstract algebra_sentence_6

Algebraic structures, with their associated homomorphisms, form mathematical categories. Abstract algebra_sentence_7

Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Abstract algebra_sentence_8

Universal algebra is a related subject that studies types of algebraic structures as single objects. Abstract algebra_sentence_9

For example, the structure of groups is a single object in universal algebra, which is called variety of groups. Abstract algebra_sentence_10

History Abstract algebra_section_0

As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra. Abstract algebra_sentence_11

Through the end of the nineteenth century, many – perhaps most – of these problems were in some way related to the theory of algebraic equations. Abstract algebra_sentence_12

Major themes include: Abstract algebra_sentence_13

Abstract algebra_unordered_list_0

Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then proceed to establish their properties. Abstract algebra_sentence_14

This creates a false impression that in algebra axioms had come first and then served as a motivation and as a basis of further study. Abstract algebra_sentence_15

The true order of historical development was almost exactly the opposite. Abstract algebra_sentence_16

For example, the hypercomplex numbers of the nineteenth century had kinematic and physical motivations but challenged comprehension. Abstract algebra_sentence_17

Most theories that are now recognized 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. Abstract algebra_sentence_18

An archetypical example of this progressive synthesis can be seen in the history of group theory. Abstract algebra_sentence_19

Early group theory Abstract algebra_section_1

There were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, and geometry. Abstract algebra_sentence_20

Leonhard Euler considered algebraic operations on numbers modulo an integer—modular arithmetic—in his generalization of Fermat's little theorem. Abstract algebra_sentence_21

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. Abstract algebra_sentence_22

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. Abstract algebra_sentence_23

In 1870, Leopold Kronecker gave a definition of an abelian group in the context of ideal class groups of a number field, generalizing Gauss's work; but it appears he did not tie his definition with previous work on groups, particularly permutation groups. Abstract algebra_sentence_24

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). Abstract algebra_sentence_25

Permutations were studied by Joseph-Louis Lagrange in his 1770 paper Réflexions sur la résolution algébrique des équations (Thoughts on the algebraic solution of equations) devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Abstract algebra_sentence_26

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. Abstract algebra_sentence_27

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. Abstract algebra_sentence_28

However, he did not consider composition of permutations. Abstract algebra_sentence_29

Serendipitously, the first edition of Edward Waring's Meditationes Algebraicae (Meditations on Algebra) appeared in the same year, with an expanded version published in 1782. Abstract algebra_sentence_30

Waring proved the fundamental theorem of symmetric polynomials, and specially considered the relation between the roots of a quartic equation and its resolvent cubic. Abstract algebra_sentence_31

Mémoire sur la résolution des équations (Memoire on the Solving of Equations) 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. Abstract algebra_sentence_32

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. Abstract algebra_sentence_33

His goal was to establish the impossibility of an algebraic solution to a general algebraic equation of degree greater than four. Abstract algebra_sentence_34

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 Abstract algebra_sentence_35

However, he got by without formalizing the concept of a group, or even of a permutation group. Abstract algebra_sentence_36

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 is now called the closure property of a group of permutations, which he expressed as Abstract algebra_sentence_37

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. Abstract algebra_sentence_38

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. Abstract algebra_sentence_39

The abstract notion of a group appeared for the first time in Arthur Cayley's papers in 1854. Abstract algebra_sentence_40

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. Abstract algebra_sentence_41

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. Abstract algebra_sentence_42

Modern algebra Abstract algebra_section_2

The end of the 19th and the beginning of the 20th century saw a shift in the methodology of mathematics. Abstract algebra_sentence_43

Abstract algebra emerged around the start of the 20th century, under the name modern algebra. Abstract algebra_sentence_44

Its study was part of the drive for more intellectual rigor in mathematics. Abstract algebra_sentence_45

Initially, the assumptions in classical algebra, on which the whole of mathematics (and major parts of the natural sciences) depend, took the form of axiomatic systems. Abstract algebra_sentence_46

No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Abstract algebra_sentence_47

Formal definitions of certain algebraic structures began to emerge in the 19th century. Abstract algebra_sentence_48

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. Abstract algebra_sentence_49

Questions of structure and classification of various mathematical objects came to forefront. Abstract algebra_sentence_50

These processes were occurring throughout all of mathematics, but became especially pronounced in algebra. Abstract algebra_sentence_51

Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such as groups, rings, and fields. Abstract algebra_sentence_52

Hence such things as group theory and ring theory took their places in pure mathematics. Abstract algebra_sentence_53

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. Abstract algebra_sentence_54

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. Abstract algebra_sentence_55

Basic concepts Abstract algebra_section_3

Main article: Algebraic structure Abstract algebra_sentence_56

By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. Abstract algebra_sentence_57

For instance, almost all systems studied are sets, to which the theorems of set theory apply. Abstract algebra_sentence_58

Those sets that have a certain binary operation defined on them form magmas, to which the concepts concerning magmas, as well those concerning sets, apply. Abstract algebra_sentence_59

We can add additional constraints on the algebraic structure, such as associativity (to form semigroups); identity, and inverses (to form groups); and other more complex structures. Abstract algebra_sentence_60

With additional structure, more theorems could be proved, but the generality is reduced. Abstract algebra_sentence_61

The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of group theory may be used when studying rings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. Abstract algebra_sentence_62

In general there is a balance between the amount of generality and the richness of the theory: more general structures have usually fewer nontrivial theorems and fewer applications. Abstract algebra_sentence_63

Examples of algebraic structures with a single binary operation are: Abstract algebra_sentence_64

Abstract algebra_unordered_list_1

Examples involving several operations include: Abstract algebra_sentence_65

Applications Abstract algebra_section_4

Because of its generality, abstract algebra is used in many fields of mathematics and science. Abstract algebra_sentence_66

For instance, algebraic topology uses algebraic objects to study topologies. Abstract algebra_sentence_67

The Poincaré conjecture, proved in 2003, asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Abstract algebra_sentence_68

Algebraic number theory studies various number rings that generalize the set of integers. Abstract algebra_sentence_69

Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem. Abstract algebra_sentence_70

In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. Abstract algebra_sentence_71

In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system. Abstract algebra_sentence_72

The groups that describe those symmetries are Lie groups, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of force carriers in a theory is equal to the dimension of the Lie algebra, and these bosons interact with the force they mediate if the Lie algebra is nonabelian. Abstract algebra_sentence_73

See also Abstract algebra_section_5

Main article: List of abstract algebra topics Abstract algebra_sentence_74

Abstract algebra_unordered_list_2

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Abstract algebra.