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 fields was used implicitly by Niels Henrik Abel and Evariste Galois 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 are indeed field of polynomials in modern terms.

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

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 of groups and fields in great details during 1928-1942.

Elementary introduction

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.

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

Generalisation and related topics

See Ring, Vector space, finite field.