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Please refer to Glossary Of Field Theory for some basic definitions in field theory. HISTORY The concept of ''field'' was used implicitly by Niels Henrik Abel and Évariste Galois in their work on the solvability of equations. In 1871, Richard Dedekind , called a set of real or complex numbers which is closed under the four arithmetic operations a "field". In 1881, Leopold Kronecker defined what he called a "domain of rationality", which is indeed a field of polynomials in modern terms. In 1893, Heinrich Weber gave the first clear definition of an abstract field. In 1910 , Perfect Field and the Transcendence Degree of an Field Extension . Galois, who did not have the term "field" in mind, is honored to be the first mathematician linking Group Theory and field theory. Galois Theory is named after him. However it was Emil Artin 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 Rational Numbers , Real Numbers , and Complex Number s. In particular, the usual rules of Associativity , Commutativity and Distributivity 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 '' Division Ring '' or sometimes a ''skew field'', but also ''non-commutative field'' is still widely used. However, other languages have retained the old usage. In French , division rings are called ''corps'' (literally, ''body''). There is no single word for field; they are simply called ''corps commutatif''. The German word for ''body'' is ''Körper'' and this word is used to denote fields; hence the use of the Blackboard Bold to denote a field. 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 Algebraically Closed Field is a field in which every polynomial has a root. For instance, the field of Algebraic Number s is the algebraic closure of the field of Rational Number s and the field of Complex Number s is the algebraic closure of the field of Real Number s. The concept of a field is of use, for example, in defining Vector s and Matrices , two structures in Linear Algebra whose components can be elements of an arbitrary field. Finite Field s are used in Coding Theory . Again algebraic extension is an important tool. Binary Field s, fields with Characteristic 2, are useful in Computer Science . 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 |