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Please refer to Glossary of field theory for some basic definitions in field theory.
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 definition 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.
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 fieldIn abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theos are used in coding theoryCoding theory deals with the properties of codes and thus with their fitness for an specific application. The aim of coding theory is to find codes which transmit fast, contain many valid code words and can correct or at least detect many errors. These ai. Again algebraic extension is an important tool. Binary field s, fields with characteristicIn mathematics, the characteristic of a ring R with identity element 1 is defined to be the smallest positive integer n such that n''1 0 (where n''1 is defined as 1 +. 1 with n summands). If no such n exists, we say that the characteristic of R is 0. 2, are useful in computer scienceIn its most general sense, computer science CS or compsci is the study of computation and information processing, both in hardware and in software. Introduction Computer science encomposses a variety of topics relating to computation, ranging from abstrac. They are usually studied as an exceptional case in finite field theory because addition and subtraction are the same operation.See RingIn ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. History See Ring theory Definition and notation A ring is an a, Vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for, finite fieldIn abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theo.
Field theoryField theory (mathematics), the theory of the algebraic concept of field. Field theory (physics), a physical theory which employs fields in the physical sense.