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Field theory (mathematics)

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Field theory (mathematics)

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 a field extension of the field of rational numbers in modern terms.[1]

In 1893, Heinrich M. Weber gave the first clear definition of an abstract field.

In 1910 Ernst Steinitz published the influential paper Algebraische Theorie der Körper (German: Algebraic Theory of Fields). In this paper he axiomatically studied the properties of fields and defined many important field theoretic concepts like prime field, perfect field and the transcendence degree of a 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 division algebra or sometimes a skew field. Also non-commutative field is still widely used. In French, 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 German word for body is Körper and this word is used to denote fields; hence the use of the blackboard bold $\mathbb K$ to denote a field.

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 Galois theory 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 algebraic closure, the separable closure, the cyclic closure 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[t] has a root in K", then a P-closure of k is just an algebraic closure 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 vectors and matrices, two structures in linear algebra whose components can be elements of an arbitrary field.

Finite fields are used in number theory, Galois theory and coding theory, and again algebraic extension is an important tool.

Binary fields, fields of characteristic 2, are useful in computer science.

References

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