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'''Field theory''' is a branch of ] which studies the properties of ]s. A field is a mathematical entity for which addition, subtraction, multiplication and division are ].
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Please refer to ] for some basic definitions in field theory.

==History==
The concept of ''field'' was used implicitly by ] and ] in their work on the solvability of equations.

In 1871, ], called a set of real or complex numbers which is closed under the four arithmetic operations a "field".

In 1881, ] defined what he called a "domain of rationality", which is indeed a field of polynomials in modern terms.

In 1893, ] gave the first clear definition of an abstract field.

In 1910 ] published the very influential paper ''Algebraische Theorie der Körper'' (]: Algebraic Theory of Fields). In this paper he axiomatically studies the properties of fields and defines many important field theoretic concepts like ], ] and the ] of an ].

Galois, who did not have the term "field" in mind, is honored to be the first mathematician linking ] and field theory. ] is named after him. However it was ] who first developed the relationship between groups and fields in great detail during 1928-1942.

==Introduction==

Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the ]s, ]s, and ]s. In particular, the usual rules of ], ] and ] hold. Fields also appear in many other areas of mathematics; see the examples below.

When abstract algebra was first being developed, the definition of a field usually did not include commutativity of multiplication, and what we today call a field would have been called either a ''commutative field'' or a ''rational domain''. In contemporary usage, a field is always commutative. A structure which satisfies all the properties of a field except possibly for commutativity, is today called a '']'' or sometimes a ''skew field'', but also ''non-commutative field'' is still widely used. However, other languages have retained the old usage. In ], division rings are called ''corps'' (literally, ''body''). There is no single word for field; they are simply called ''corps commutatif''. The ] word for ''body'' is ''Körper'' and this word is used to denote fields; hence the use of the ] <math>\mathbb K</math> to denote a field. <!-- see talk page for why other languages are not included. -->

The concept of fields was first used to prove that there is no general formula for the roots of real polynomials of degree higher than 4.

The central concept of Galois theory is the algebraic extension of an underlying field. It is simply the smallest field containing the underlying field and a root of a polynomial. An ] is a field in which every polynomial has a root. For instance, the field of ]s is the algebraic closure of the field of ]s and the field of ]s is the algebraic closure of the field of ]s.

The concept of a field is of use, for example, in defining ]s and ], two structures in ] whose components can be elements of an arbitrary field.

]s are used in ]. Again algebraic extension is an important tool.

]s, fields with ] 2, are useful in ]. They are usually studied as an exceptional case in finite field theory because addition and subtraction are the same operation.

== Some useful theorems ==

*]
*]

==See also==
⚫ * ]
* ]
* ]

==References==
* {{cite book | author=R.B.J.T. Allenby | title=Rings, Fields and Groups|publisher= Butterworth-Heinemann | year=1991 | id=ISBN 0-3405-4440-6}}
* {{cite book | author=T.S. Blyth and E.F. Robertson| title=Groups, rings and fields: Algebra through practice, Book 3| publisher= Cambridge Univeristy Press| year=1985| id=ISBN 0-521-27288-2}}
* {{cite book | author=T.S. Blyth and E.F. Robertson| title=Rings, fields and modules: Algebra through practice, Book 6| publisher= Cambridge Univeristy Press| year=1985| id=ISBN 0-521-27291-2}}

==External links==
*

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Latest revision as of 21:12, 6 May 2024

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