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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 '']'' 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 a ].

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==

]s 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 ''division algebra'' or sometimes a ''skew field''. Also ''non-commutative field'' is still widely used. In ], fields are called ''corps'' (literally, ''body''), generally regardless of their commutativity. When necessary, a (commutative) field is called ''corps commutatif'' and a skew field ''corps gauche''. 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 (implicitly) used to prove that there is no general formula expressing in terms of radicals the roots of a polynomial with rational coefficients of degree 5 or higher.

==Extensions of a field==

An extension of a field ''k'' is just a field ''K'' containing ''k'' as a subfield. One distinguishes between extensions having various qualities. For example, an extension ''K'' of a field ''k'' is called ''algebraic'', if every element of ''K'' is a root of some polynomial with coefficients in ''k''. Otherwise, the extension is called ''transcendental''.

The aim of ] is the study of ''algebraic extensions'' of a field.

==Closures of a field==
Given a field ''k'', various kinds of closures of ''k'' may be introduced. For example, the ], the ], the ] et cetera. The idea is always the same: If ''P'' is a property of fields, then a ''P''-closure of ''k'' is a field ''K'' containing ''k'', having property ''P'', and which is minimal in the sense that no proper subfield of ''K'' that contains ''k'' has property ''P''.
For example if we take ''P(K)'' to be the property "every nonconstant polynomial ''f'' in ''K'' has a root in ''K''", then a ''P''-closure of ''k'' is just an ] of ''k''.
In general, if ''P''-closures exist for some property ''P'' and field ''k'', they are all isomorphic. However, there is in general no preferable isomorphism between two closures.

==Applications of field theory==

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 ], ] and ], and again algebraic extension is an important tool.

]s, fields of ] 2, are useful in ].

== 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 0-340-54440-6}}
* {{cite book | author=T.S. Blyth and E.F. Robertson| title=Groups, rings and fields: Algebra through practice, Book 3| publisher= Cambridge University 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 University Press| year=1985| id=ISBN 0-521-27291-2}}


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