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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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...
Please refer to Glossary of field theory for some basic definitions in field theory. Field theory is the branch of mathematics in which fields are studied. ...
History
The concept of field was used implicitly by Niels Henrik Abel and Évariste Galois in their work on the solvability of equations. Niels Henrik Abel (August 5, 1802–April 6, 1829), Norwegian mathematician, was born in Finnøy. ...
Galois at the age of fifteen from the pencil of a classmate. ...
In 1871, Richard Dedekind, called a set of real or complex numbers which is closed under the four arithmetic operations a "field". Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ...
In 1881, Leopold Kronecker defined what he called a "domain of rationality", which is indeed a field of polynomials in modern terms. Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ...
In 1893, Heinrich Weber gave the first clear definition of an abstract field. Heinrich Martin Weber (1842–1913) was a German mathematician who specialized in algebra and number theory. ...
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. Ernst Steinitz (June 13, 1871 – September 29, 1928) was an German mathematician. ...
In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ...
In mathematics, a separable extension of a field K is a field L containing K that can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K. In that case, each β in L has a separable...
In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the size of the extension. ...
In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ...
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. Group theory is that branch of mathematics concerned with the study of groups. ...
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ...
Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ...
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. In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematics, associativity is a property that a binary operation can have. ...
In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...
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. In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠1 and such that every non-zero element a has a multiplicative inverse (i. ...
An example of blackboard bold letters. ...
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. In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ...
In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ...
In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational 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. In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. ...
Finite fields are used in number theory, Galois theory and coding theory, and again algebraic extension is an important tool. In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...
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In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ...
Coding theory is a branch of mathematics and computer science dealing with the error-prone process of transmitting data across noisy channels, via clever means, so that a large number of errors that occur can be corrected. ...
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. In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...
Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...
Some useful theorems In mathematics, a primitive element for an extension of fields L/K is an element ζ of L such that L = K(ζ), or in other words such that L is generated by ζ over K. This means that every element of L can be written as a quotient of...
See also In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...
In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...
In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...
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 University Press. ISBN 0-521-27288-2.
- T.S. Blyth and E.F. Robertson (1985). Rings, fields and modules: Algebra through practice, Book 6. Cambridge University Press. ISBN 0-521-27291-2.
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