Glossary of field theory
   HOME

TheInfoList



OR:

Field theory is the branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
in which
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
s are studied. This is a glossary of some terms of the subject. (See
field theory (physics) In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each point on ...
for the unrelated field theories in physics.)


Definition of a field

A field is a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
(''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 In mathematics, 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 the order in which they are written. That is, the group operation is commut ...
under multiplication; this group is typically denoted by ''F''×; The
ring of polynomials In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
in the variable ''x'' with coefficients in ''F'' is denoted by ''F'' 'x''


Basic definitions

; Characteristic : The ''characteristic'' of the field ''F'' is the smallest positive
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. For example, the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s and the ''p''-adic numbers have characteristic 0, while the finite field Z''p'' where ''p'' is prime has characteristic ''p''. ; Subfield : A ''subfield'' of a field ''F'' is a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of ''F'' which is closed under the field operation + and * of ''F'' and which, with these operations, forms itself a field. ;
Prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive iden ...
: The ''prime field'' of the field ''F'' is the unique smallest subfield of ''F''. ;
Extension field In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
: 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 In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory — ...
: Given an extension ''E''/''F'', the field ''E'' can be considered as a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over the field ''F'', and the
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
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 In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
: If an element α of an extension field ''E'' over ''F'' is the
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
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 In field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization ...
. ;
Splitting field In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors. Definition A splitting field of a poly ...
: A field extension generated by the complete factorisation of a polynomial. ;
Normal extension In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic ext ...
: A field extension generated by the complete factorisation of a set of polynomials. ;
Separable extension In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynom ...
: An extension generated by roots of
separable polynomial In mathematics, a polynomial ''P''(''X'') over a given field ''K'' is separable if its roots are distinct in an algebraic closure of ''K'', that is, the number of distinct roots is equal to the degree of the polynomial. This concept is closely ...
s. ;
Perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' is ...
: A field such that every finite extension is separable. All fields of characteristic zero, and all finite fields, are perfect. ;
Imperfect degree The imperfect (abbreviated ) is a verb form that combines past tense (reference to a past time) and imperfective aspect (reference to a continuing or repeated event or state). It can have meanings similar to the English "was walking" or "used to w ...
: Let ''F'' be a field of characteristic ''p''>0; then ''F''''p'' is a subfield. The degree 'F'':''F''''p''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 ''p''n.Fried & Jarden (2008) p.45 ;
Algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
: 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 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. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1 ...
: 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 In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...
: The number of algebraically independent transcendental elements in a field extension. It is used to define the
dimension of an algebraic variety In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutati ...
.


Homomorphisms

; Field homomorphism : A ''field homomorphism'' between two fields ''E'' and ''F'' is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
::''f'' : ''E'' → ''F'' :such that, for all ''x'', ''y'' in ''E'', ::''f''(''x'' + ''y'') = ''f''(''x'') + ''f''(''y'') ::''f''(''xy'') = ''f''(''x'') ''f''(''y'') ::''f''(1) = 1. :These properties imply that , for ''x'' in ''E'' with , and that ''f'' is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. Fields, together with these homomorphisms, form a
category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...
. Two fields ''E'' and ''F'' are called isomorphic if there exists a
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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 In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
.


Types of fields

;
Finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
: A field with finitely many elements. Aka Galois 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. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field ...
: A field with a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive) ...
compatible with its operations. ;
Rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s ;
Real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s ;
Complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s ;
Number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
: Finite extension of the field of rational numbers. ;
Algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
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 is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
. ;
Quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 an ...
: A degree-two extension of the rational numbers. ;
Cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of th ...
: An extension of the rational numbers generated by a
root of unity In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
. ;
Totally real field In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyno ...
: A number field generated by a root of a polynomial, having all its roots real numbers. ;
Formally real field In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field. Alternative definitions The definition given above is ...
;
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. Def ...
;
Global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...
: A number field or a function field of one variable over a finite field. ;
Local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
: A completion of some global field ( w.r.t. a prime of the integer ring). ;
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 (such as the ''p''-adic numbers). Constructio ...
: A field complete w.r.t. to some valuation. ;
Pseudo algebraically closed field In mathematics, a field (mathematics), field K is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.Fried & Jarden (2008) p.218 Formulation A ...
: A field in which every variety has a
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
.Fried & Jarden (2008) p.214 ; Henselian field : A field satisfying Hensel lemma w.r.t. some valuation. A generalization of complete fields. ;
Hilbertian field In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field ''K'', by allowed operations that are in a definite sense 'unlikely'. The two fundam ...
: A field satisfying
Hilbert's irreducibility theorem In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization o ...
: formally, one for which the
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
is not thin in the sense of Serre.Serre (1992) p.19Schinzel (2000) p.298 ; Kroneckerian field: A totally real algebraic number field or a totally imaginary quadratic extension of a totally real field.Schinzel (2000) p.5 ;
CM-field In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field. The abbreviation "CM" was introduced by . Formal definition A number field '' ...
or J-field: An algebraic number field which is a totally imaginary quadratic extension of a totally real field. ;
Linked field In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property. Linked quaternion algebras Let ''F'' be a field of characteristic not equal to 2. Let ''A'' = (''a''1,''a''2) and ''B ...
: A field over which no biquaternion algebra is a
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...
.Lam (2005) p.342 ; Frobenius field: A
pseudo algebraically closed field In mathematics, a field (mathematics), field K is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.Fried & Jarden (2008) p.218 Formulation A ...
whose
absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' tha ...
has the embedding property.Fried & Jarden (2008) p.564


Field extensions

Let ''E''/''F'' be a field extension. ;
Algebraic extension In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
: An extension in which every element of ''E'' is algebraic over ''F''. ;
Simple extension In field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization ...
: An extension which is generated by a single element, called a primitive element, or generating element. The
primitive element theorem In field theory, the primitive element theorem is a result characterizing the finite degree field extensions that can be generated by a single element. Such a generating element is called a primitive element of the field extension, and the exten ...
classifies such extensions. ;
Normal extension In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic ext ...
: An extension that splits a family of polynomials: every root of the minimal polynomial of an element of ''E'' over ''F'' is also in ''E''. ;
Separable extension In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynom ...
: An algebraic extension in which the minimal polynomial of every element of ''E'' over ''F'' is a
separable polynomial In mathematics, a polynomial ''P''(''X'') over a given field ''K'' is separable if its roots are distinct in an algebraic closure of ''K'', that is, the number of distinct roots is equal to the degree of the polynomial. This concept is closely ...
, that is, has distinct roots.Fried & Jarden (2008) p.28 ;
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...
: A normal, separable field extension. ;
Primary extension In field theory, a branch of algebra, a primary extension ''L'' of ''K'' is a field extension such that the algebraic closure of ''K'' in ''L'' is purely inseparable over ''K''.Fried & Jarden (2008) p.44 Properties * An extension ''L''/''K'' is ...
: An extension ''E''/''F'' such that the algebraic closure of ''F'' in ''E'' is purely inseparable over ''F''; equivalently, ''E'' is
linearly disjoint In mathematics, algebras ''A'', ''B'' over a field ''k'' inside some field extension \Omega of ''k'' are said to be linearly disjoint over ''k'' if the following equivalent conditions are met: *(i) The map A \otimes_k B \to AB induced by (x, y) \ma ...
from the
separable 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. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of ''F''.Fried & Jarden (2008) p.44 ;
Purely transcendental extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
: An extension ''E''/''F'' in which every element of ''E'' not in ''F'' is transcendental over ''F''. ; Regular extension : An extension ''E''/''F'' such that ''E'' is separable over ''F'' and ''F'' is algebraically closed in ''E''. ; Simple radical extension: A
simple extension In field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization ...
''E''/''F'' generated by a single element α satisfying \alpha^n = b for an element ''b'' of ''F''. In characteristic ''p'', we also take an extension by a root of an Artin–Schreier polynomial to be a simple radical extension.Roman (2007) p.273 ;
Radical extension In mathematics and more specifically in field theory, a radical extension of a field ''K'' is an extension of ''K'' that is obtained by adjoining a sequence of ''n''th roots of elements. Definition A simple radical extension is a simple extensi ...
: A tower F = F_0 < F_1 < \cdots < F_k = E where each extension F_i / F_ is a simple radical extension. ; Self-regular extension : An extension ''E''/''F'' such that ''E'' ⊗''F'' ''E'' is an integral domain. ; Totally transcendental extension: An extension ''E''/''F'' such that ''F'' is algebraically closed in ''F''. ; Distinguished class: A class ''C'' of field extensions with the three propertiesLang (2002) p.228 :# If ''E'' is a C-extension of ''F'' and ''F'' is a C-extension of ''K'' then ''E'' is a C-extension of ''K''. :# If ''E'' and ''F'' are C-extensions of ''K'' in a common overfield ''M'', then the
compositum In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subf ...
''EF'' is a C-extension of ''K''. :# If ''E'' is a C-extension of ''F'' and ''E'' > ''K'' > ''F'' then ''E'' is a C-extension of ''K''.


Galois theory

;
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...
: A normal, separable field extension. ;
Galois group In mathematics, in the area of abstract 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 study of field extensions and their relationship to the pol ...
: The
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
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 group In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...
s. ;
Kummer theory In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of ''n''th roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer aro ...
: The Galois theory of taking ''n''-th roots, given enough
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
. It includes the general theory of
quadratic extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s. ;
Artin–Schreier theory In mathematics, Artin–Schreier theory is a branch of Galois theory, specifically a positive characteristic analogue of Kummer theory, for Galois extensions of degree equal to the characteristic ''p''. introduced Artin–Schreier theory for ex ...
: Covers an exceptional case of Kummer theory, in characteristic ''p''. ;
Normal basis In mathematics, specifically the algebraic theory of fields, a normal basis 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 f ...
: 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 In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfie ...
: A different foundational piece of algebra, including the
compositum In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subf ...
operation (
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two top ...
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 In mathematics, differential Galois theory studies the Galois groups of differential equations. Overview Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential field ...
: The subject in which symmetry groups of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations when
Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Life and career Marius Sophu ...
founded the theory of
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s. It has not, probably, reached definitive form. ;
Grothendieck's Galois theory In mathematics, Grothendieck's Galois theory is an 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. It provides, in ...
: A very abstract approach from
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, introduced to study the analogue of the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
.


References

* * * * * * * * * {{DEFAULTSORT:Glossary Of Field Theory Field theory * Wikipedia glossaries using description lists