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Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined.

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 Ernst Steinitz published the very influential paper Algebraische Theorie der Körper (German: Algebraic Theory of Fields). In this paper he axiomatically studies the properties of fields and defines many important field theoretic concepts like prime field, 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 numbers. 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 ${\displaystyle \mathbb {K} }$ 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 numbers is the algebraic closure of the field of rational numbers and the field of complex numbers is the algebraic closure of the field of real numbers.

The concept of a field is of use, for example, in defining vectors and matrices, two structures in linear algebra whose components can be elements of an arbitrary field.

Finite fields are used in coding theory. Again algebraic extension is an important tool.

Binary fields, 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

• Isomorphism extension theorem
• Primitive element theorem

See also

• Ring
• Vector space
• finite field

References

• R.B.J.T. Allenby (1991). Rings, Fields and Groups. Butterworth-Heinemann. ISBN 0-3405-4440-6.
• T.S. Blyth and E.F. Robertson (1985). Groups, rings and fields: Algebra through practice, Book 3. Cambridge Univeristy Press. ISBN 0-521-27288-2.
• T.S. Blyth and E.F. Robertson (1985). Rings, fields and modules: Algebra through practice, Book 6. Cambridge Univeristy Press. ISBN 0-521-27291-2.

External links

• Field Theory Q&A

• Field theory