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| '''Field theory''' is a branch of ] that studies the properties of ]s. A field is a mathematical entity for which addition, subtraction, multiplication and division are ]. | |||
| 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 a ] of the ] in modern terms.<ref>{{cite book | title=Galois Theory | volume=106 | series=Pure and Applied Mathematics | first=David A. | last=Cox | edition=2nd | publisher=John Wiley & Sons | year=2012 | isbn=1118218426 | page=348 }}</ref> | |||
| In 1893, ] gave the first clear definition of an abstract field. | |||
| In 1910 ] published the influential paper ''Algebraische Theorie der Körper'' (]: Algebraic Theory of Fields). In this paper he axiomatically studied the properties of fields and defined 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. | |||
| ==Branches of field theory== | |||
| The following list is not complete. | |||
| ===Galois theory=== | |||
| {{main|Galois theory}} | |||
| '''Galois theory''', named after ], provides a connection between ] and ]. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood. | |||
| Originally, Galois used ]s to describe how the various ]s of a given ] equation are related to each other. The modern approach to Galois theory, developed by ], ] and ], among others, involves studying ]s of ]s. | |||
| === Algebraic number theory === | |||
| {{main|Algebraic number theory}} | |||
| An ] (or simply '''number field''') ''F'' is a finite (and hence ]) ] of the ] of ]s '''Q'''. Thus ''F'' is a field that contains '''Q''' and has finite ] when considered as a ] over '''Q'''. | |||
| The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of ]. | |||
| === Finite field theory === | |||
| {{main|finite field}} | |||
| In ], a '''finite field''' or '''Galois field''' (so named in honor of ]) is a ] that contains a finite number of ], called its ''order'' (the ''size'' of the underlying set). As with any field, a finite field is a ] on which the operations of ] multiplication, addition, subtraction and division (by anything except zero) have been defined. Common, but not the only, examples of finite fields are given by the '''integers modulo a prime''', that is, the ] where ''n'' is a prime number, such as ℤ/3ℤ or ℤ/7ℤ. | |||
| Finite fields are fundamental in a number of areas of mathematics and computer science, including ], ], ], ], ] and ]. | |||
| ==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== | |||
| {{reflist}} | |||
| * {{cite book | first=R.B.J.T. | last=Allenby | title=Rings, Fields and Groups|publisher= Butterworth-Heinemann | year=1991 | id=ISBN 0-340-54440-6}} | |||
| * {{cite book | first1=T.S. | last1=Blyth | first2=E.F. | last2=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 | first1=T.S. | last1=Blyth | first2=E.F. | last2=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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