Glossary of field theory
Encyclopedia
Field theory
is the branch of mathematics
in which field
s are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.)
(F,+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division.
The non-zero elements of a field F form an abelian group
under multiplication; this group is typically denoted by F×;
The ring of polynomials
in the variable x with coefficients in F is denoted by F[x].
: The characteristic of the field F is the smallest positive integer
n such that n·1 = 0; here n·1 stands for n summands 1 + 1 + 1 + ... + 1. If no such n exists, we say the characteristic is zero. Every non-zero characteristic is a prime number
. For example, the rational number
s, the real number
s and the p-adic numbers have characteristic 0, while the finite field Zp has characteristic p.
Subfield : A subfield of a field F is a subset
of F which is closed under the field operation + and * of F and which, with these operations, forms itself a field.
Prime field : The prime field of the field F is the unique smallest subfield of F.
Extension field
: If F is a subfield of E then E is an extension field of F. We then also say that E/F is a field extension.
Degree of an extension
: Given an extension E/F, the field E can be considered as a vector space
over the field F, and the dimension
of this vector space is the degree of the extension, denoted by [E : F].
Finite extension : A finite extension is a field extension whose degree is finite.
Algebraic extension
: If an element α of an extension field E over F is the root
of a non-zero polynomial in F[x], then α is algebraic over F. If every element of E is algebraic over F, then E/F is an algebraic extension.
Generating set : Given a field extension E/F and a subset S of E, we write F(S) for the smallest subfield of E that contains both F and S. It consists of all the elements of E that can be obtained by repeatedly using the operations +,-,*,/ on the elements of F and S. If E = F(S) we say that E is generated by S over F.
Primitive element : An element α of an extension field E over a field F is called a primitive element if E=F(α), the smallest extension field containing α. Such an extension is called a simple extension
.
Splitting field
: A field extension generated by the complete factorisation of a polynomial.
Normal extension
: A field extension generated by the complete factorisation of a set of polynomials.
Separable extension
: An extension generated by roots of separable polynomial
s.
Perfect field
: A field such that every finite extension is separable. All fields of characteristic zero, and all finite fields, are perfect.
Imperfect degree : Let F be a field of characteristic p>0; then Fp is a subfield. The degree [F:Fp] is called the imperfect degree of F. The field F is perfect if and only if its imperfect degree is 1. For example, if F is a function field of n variables over a finite field of characteristic p>0, then its imperfect degree is pn.
Algebraically closed field
: A field F is algebraically closed if every polynomial in F[x] has a root in F; equivalently: every polynomial in F[x] is a product of linear factors.
Algebraic closure
: An algebraic closure of a field F is an algebraic extension of F which is algebraically closed. Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F.
Transcendental : Those elements of an extension field of F that are not algebraic over F are transcendental over F.
Algebraically independent elements : Elements of an extension field of F are algebraically independent over F if they don't satisfy any non-zero polynomial equation with coefficients in F.
Transcendence degree
: The number of algebraically independent transcendental elements in a field extension. It is used to define the dimension of an algebraic variety
.
: A field with finitely many elements.
Ordered field
: A field with a total order
compatible with its operations.
Rational number
s
Real number
s
Complex number
s
Number field : Finite extension of the field of rational numbers.
Algebraic number
s : The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers. Their detailed properties are studied in algebraic number theory
.
Quadratic field
: A degree-two extension of the rational numbers.
Cyclotomic field
: An extension of the rational numbers generated by a root of unity
.
Totally real field : A number field generated by a root of a polynomial, having all its roots real numbers.
Formally real field
Real closed field
Global field
: A number field or a function field of one variable over a finite field.
Local field
: A completion of some global field (w.r.t. a prime of the integer ring).
Complete field
: A field complete w.r.t. to some valuation.
Pseudo algebraically closed field
: A field in which every variety has a rational point.
Henselian field : A field satisfying Hensel lemma w.r.t. some valuation. A generalization of complete fields.
: A normal, separable field extension.
Galois group
: The automorphism group of a Galois extension. When it is a finite extension, this is a finite group of order equal to the degree of the extension. Galois groups for infinite extensions are profinite groups.
Kummer theory
: The Galois theory of taking n-th roots, given enough roots of unity. It includes the general theory of quadratic extensions.
Artin-Schreier theory
: Covers an exceptional case of Kummer theory, in characteristic p.
Normal basis
: A basis in the vector space sense of L over K, on which the Galois group of L over K acts transitively.
Tensor product of fields
: A different foundational piece of algebra, including the compositum operation (join
of fields).
Differential Galois theory
: The subject in which symmetry groups of differential equation
s are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Lie
founded the theory of Lie group
s. It has not, probably, reached definitive form.
Grothendieck's Galois theory
: A very abstract approach from algebraic geometry
, introduced to study the analogue of the fundamental group
.
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....
is the branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
in which field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
s are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.)
Definition of a field
A field is a commutative ringCommutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
(F,+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division.
The non-zero elements of a field F form an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
under multiplication; this group is typically denoted by F×;
The ring of polynomials
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
in the variable x with coefficients in F is denoted by F[x].
Basic definitions
CharacteristicCharacteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
: The characteristic of the field F is the smallest positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
n such that n·1 = 0; here n·1 stands for n summands 1 + 1 + 1 + ... + 1. If no such n exists, we say the characteristic is zero. Every non-zero characteristic is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. For example, the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s and the p-adic numbers have characteristic 0, while the finite field Zp has characteristic p.
Subfield : A subfield of a field F is a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of F which is closed under the field operation + and * of F and which, with these operations, forms itself a field.
Prime field : The prime field of the field F is the unique smallest subfield of F.
Extension field
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
: If F is a subfield of E then E is an extension field of F. We then also say that E/F is a field extension.
Degree of an extension
Degree of a field extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — indeed in any area where fields appear prominently.-...
: Given an extension E/F, the field E can be considered as a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over the field F, and the dimension
Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...
of this vector space is the degree of the extension, denoted by [E : F].
Finite extension : A finite extension is a field extension whose degree is finite.
Algebraic extension
Algebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
: If an element α of an extension field E over F is the root
Root
In vascular plants, the root is the organ of a plant that typically lies below the surface of the soil. This is not always the case, however, since a root can also be aerial or aerating . Furthermore, a stem normally occurring below ground is not exceptional either...
of a non-zero polynomial in F[x], then α is algebraic over F. If every element of E is algebraic over F, then E/F is an algebraic extension.
Generating set : Given a field extension E/F and a subset S of E, we write F(S) for the smallest subfield of E that contains both F and S. It consists of all the elements of E that can be obtained by repeatedly using the operations +,-,*,/ on the elements of F and S. If E = F(S) we say that E is generated by S over F.
Primitive element : An element α of an extension field E over a field F is called a primitive element if E=F(α), the smallest extension field containing α. Such an extension is called a simple extension
Simple extension
In mathematics, more specifically in field theory, a simple extension is a field extension which is generated by the adjunction of a single element...
.
Splitting field
Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...
: A field extension generated by the complete factorisation of a polynomial.
Normal extension
Normal extension
In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]...
: A field extension generated by the complete factorisation of a set of polynomials.
Separable extension
Separable extension
In modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...
: An extension generated by roots of separable polynomial
Separable polynomial
In mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
s.
Perfect field
Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
: A field such that every finite extension is separable. All fields of characteristic zero, and all finite fields, are perfect.
Imperfect degree : Let F be a field of characteristic p>0; then Fp is a subfield. The degree [F:Fp] is called the imperfect degree of F. The field F is perfect if and only if its imperfect degree is 1. For example, if F is a function field of n variables over a finite field of characteristic p>0, then its imperfect degree is pn.
Algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
: A field F is algebraically closed if every polynomial in F[x] has a root in F; equivalently: every polynomial in F[x] is a product of linear factors.
Algebraic closure
Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
: An algebraic closure of a field F is an algebraic extension of F which is algebraically closed. Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F.
Transcendental : Those elements of an extension field of F that are not algebraic over F are transcendental over F.
Algebraically independent elements : Elements of an extension field of F are algebraically independent over F if they don't satisfy any non-zero polynomial equation with coefficients in F.
Transcendence degree
Transcendence degree
In abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension...
: The number of algebraically independent transcendental elements in a field extension. It is used to define the dimension of an algebraic variety
Dimension of an algebraic variety
In mathematics, the dimension of an algebraic variety V in algebraic geometry is defined, informally speaking, as the number of independent rational functions that exist on V.For example, an algebraic curve has by definition dimension 1...
.
Homomorphisms
Field homomorphism : A field homomorphism between two fields E and F is a functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
-
- f : E → F
- such that
- f(x + y) = f(x) + f(y)
- and
- f(xy) = f(x) f(y)
- for all x, y in E, as well as f(1) = 1. These properties imply that f(0) = 0, f(x-1) = f(x)-1 for x in E with x ≠ 0, and that f is injective. Fields, together with these homomorphisms, form a categoryCategory theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. Two fields E and F are called isomorphic if there exists a bijective homomorphism- f : E → F.
- The two fields are then identical for all practical purposes; however, not necessarily in a unique way. See, for example, complex conjugation.
Types of fields
Finite fieldFinite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
: A field with finitely many elements.
Ordered field
Ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
: A field with a total order
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
compatible with its operations.
Rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s
Real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s
Complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s
Number field : Finite extension of the field of rational numbers.
Algebraic number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
s : The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers. Their detailed properties are studied in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
.
Quadratic field
Quadratic field
In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields...
: A degree-two extension of the rational numbers.
Cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
: An extension of the rational numbers generated by a root of unity
Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
.
Totally real field : A number field generated by a root of a polynomial, having all its roots real numbers.
Formally real field
Formally real field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that admits an ordering which makes it an ordered field.-Alternative Definitions:...
Real closed field
Real closed field
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.-Definitions:...
Global field
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
: A number field or a function field of one variable over a finite field.
Local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
: A completion of some global field (w.r.t. a prime of the integer ring).
Complete field
Complete field
In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields .- See also :...
: A field complete w.r.t. to some valuation.
Pseudo algebraically closed field
Pseudo algebraically closed field
In mathematics, a field K is pseudo algebraically closed if it satisfies certain properties which hold for any algebraically closed field.-Formulation:...
: A field in which every variety has a rational point.
Henselian field : A field satisfying Hensel lemma w.r.t. some valuation. A generalization of complete fields.
Galois theory
Galois extensionGalois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
: A normal, separable field extension.
Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
: The automorphism group of a Galois extension. When it is a finite extension, this is a finite group of order equal to the degree of the extension. Galois groups for infinite extensions are profinite groups.
Kummer theory
Kummer theory
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's last...
: The Galois theory of taking n-th roots, given enough roots of unity. It includes the general theory of quadratic extensions.
Artin-Schreier theory
Artin-Schreier theory
In mathematics, Artin–Schreier theory is a branch of Galois theory, and more specifically is a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic p...
: Covers an exceptional case of Kummer theory, in characteristic p.
Normal basis
Normal basis
In mathematics, a normal basis in field theory is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis...
: A basis in the vector space sense of L over K, on which the Galois group of L over K acts transitively.
Tensor product of fields
Tensor product of fields
In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring is never itself a field. On the other hand it is often required to 'join' two fields K and L, either in cases where K and L are given as subfields of a larger field M, or when K...
: A different foundational piece of algebra, including the compositum operation (join
Join
Join may refer to:* Join , to include additional counts or additional defendants on an indictment* Join , a least upper bound of set orders in lattice theory* Join , a type of binary operator...
of fields).
Extensions of Galois theory
Inverse problem of Galois theory : Given a group G, find an extension of the rational number or other field with G as Galois group.Differential Galois theory
Differential Galois theory
In mathematics, differential Galois theory studies the Galois groups of differential equations.Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of...
: The subject in which symmetry groups of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
founded the theory of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s. It has not, probably, reached definitive form.
Grothendieck's Galois theory
Grothendieck's Galois theory
In mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry...
: A very abstract approach from algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, introduced to study the analogue of the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
.