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abstract algebra
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{{about|the branch of mathematics|the Swedish band|Abstrakt Algebra}}{{Redirect|Modern algebra|van der Waerden's book|Moderne Algebra}}File:Rubik's cube v2.svg|thumb | alt=Picture of a Rubik's Cube | The permutations of Rubik's Cube form a group, a fundamental concept within abstract algebra.]]In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures.Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called variety of groups.

## History

As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra. Through the end of the nineteenth century, many{{snd}}perhaps most{{snd}}of these problems were in some way related to the theory of algebraic equations. Major themes include:
• Solving of systems of linear equations, which led to linear algebra
• Attempts to find formulas for solutions of general polynomial equations of higher degree that resulted in discovery of groups as abstract manifestations of symmetry
• Arithmetical investigations of quadratic and higher degree forms and diophantine equations, that directly produced the notions of a ring and ideal.
Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then proceed to establish their properties. 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. The true order of historical development was almost exactly the opposite. For example, the hypercomplex numbers of the nineteenth century had kinematic and physical motivations but challenged comprehension. 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. An archetypical example of this progressive synthesis can be seen in the history of group theory.

### Modern algebra

The end of the 19th and the beginning of the 20th century saw a tremendous shift in the methodology of mathematics. Abstract algebra emerged around the start of the 20th century, under the name modern algebra. Its study was part of the drive for more intellectual rigor in mathematics. 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. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain algebraic structures began to emerge in the 19th century. 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. Hence such things as group theory and ring theory took their places in pure mathematics. 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. 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 1930â€“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.

## Basic concepts

By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied are sets, to which the theorems of set theory apply. 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. 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. With additional structure, more theorems could be proved, but the generality is reduced. 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. 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.Examples of algebraic structures with a single binary operation are: Examples involving several operations include:{{div col|colwidth=22em}} {{div col end}}

## Applications

Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance, algebraic topology uses algebraic objects to study topologies. The PoincarÃ© 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. Algebraic number theory studies various number rings that generalize the set of integers. Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem.In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system. 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.{{Citation|last=Schumm|first=Bruce|title=Deep Down Things|year=2004|publisher=Johns Hopkins University Press|location=Baltimore|isbn=0-8018-7971-X}}

## Sources

• {{Citation | last1=Allenby | first1=R. B. J. T. | title=Rings, Fields and Groups | publisher=Butterworth-Heinemann | isbn=978-0-340-54440-2 | year=1991}}
• {{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | title=Algebra | publisher=Prentice Hall | isbn=978-0-89871-510-1 | year=1991}}
• {{Citation | last1=Burris | first1=Stanley N. | last2=Sankappanavar | first2=H. P. | title=A Course in Universal Algebra | origyear=1981 | url=http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html | year=1999}}
• {{Citation | last1=Gilbert | first1=Jimmie | last2=Gilbert | first2=Linda | title=Elements of Modern Algebra | publisher=Thomson Brooks/Cole | isbn=978-0-534-40264-8 | year=2005}}
• {{Citation | last1=Sethuraman | first1=B. A. | title=Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-94848-5 | year=1996}}
• {{Citation | last1=Whitehead | first1=C. | title=Guide to Abstract Algebra | edition=2nd | isbn=978-0-333-79447-0 | year=2002 | publisher=Palgrave | location=Houndmills}}
• W. Keith Nicholson (2012) Introduction to Abstract Algebra, 4th edition, John Wiley & Sons {{ISBN|978-1-118-13535-8}} .
• John R. Durbin (1992) Modern Algebra : an introduction, John Wiley & Sons

{{Algebra}}{{Areas of mathematics | state=collapsed}}{{Authority control}}

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