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In mathematics, particularly in
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, a field extension is a pair of
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
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 ''F''. For example, under the usual notions of addition and multiplication, the
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 fo ...
s are an extension field of 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 ...
s; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
, and are widely used in algebraic geometry.


Subfield

A subfield K of a
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 ...
L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L have the same zero element. For example, the field of
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 rat ...
s is a subfield of 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 ...
s, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is isomorphic to) a subfield of any field of characteristic 0. The characteristic of a subfield is the same as the characteristic of the larger field.


Extension field

If ''K'' is a subfield of ''L'', then ''L'' is an extension field or simply extension of ''K'', and this pair of fields is a field extension. Such a field extension is denoted ''L'' / ''K'' (read as "''L'' over ''K''"). If ''L'' is an extension of ''F'', which is in turn an extension of ''K'', then ''F'' is said to be an intermediate field (or intermediate extension or subextension) of ''L'' / ''K''. Given a field extension , the larger field ''L'' is a ''K''-
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 ...
. The
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of this vector space is called the degree of the extension and is denoted by 'L'' : ''K'' The degree of an extension is 1 if and only if the two fields are equal. In this case, the extension is a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. A finite extension is an extension that has a finite degree. Given two extensions and , the extension is finite if and only if both and are finite. In this case, one has : : K : Lcdot : K Given a field extension ''L'' / ''K'' and a subset ''S'' of ''L'', there is a smallest subfield of ''L'' that contains ''K'' and ''S''. It is the intersection of all subfields of ''L'' that contain ''K'' and ''S'', and is denoted by ''K''(''S''). One says that ''K''(''S'') is the field ''generated'' by ''S'' over ''K'', and that ''S'' is a
generating set In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
of ''K''(''S'') over ''K''. When S=\ is finite, one writes K(x_1, \ldots, x_n) instead of K(\), and one says that ''K''(''S'') is over ''K''. If ''S'' consists of a single element ''s'', the 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 o ...
and ''s'' is called a primitive element of the extension. An extension field of the form is often said to result from the ' of ''S'' to ''K''. In characteristic 0, every finite extension is a simple extension. This is 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 extens ...
, which does not hold true for fields of non-zero characteristic. If a simple extension is not finite, the field ''K''(''s'') is isomorphic to the field of
rational fraction In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmetic fractions. A rationa ...
s in ''s'' over ''K''.


Caveats

The notation ''L'' / ''K'' is purely formal and does not imply the formation of a
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
or quotient group or any other kind of division. Instead the slash expresses the word "over". In some literature the notation ''L'':''K'' is used. It is often desirable to talk about field extensions in situations where the small field is not actually contained in the larger one, but is naturally embedded. For this purpose, one abstractly defines a field extension as an injective
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...
between two fields. ''Every'' non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper ideals, so field extensions are precisely the morphisms in the
category of fields In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings i ...
. Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.


Examples

The field of complex numbers \Complex is an extension field of the field of
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 ...
s \R, and \R in turn is an extension field of the field of rational numbers \Q. Clearly then, \Complex/\Q is also a field extension. We have Complex:\R=2 because \ is a basis, so the extension \Complex/\R is finite. This is a simple extension because \Complex = \R(i). R:\Q=\mathfrak c (the cardinality of the continuum), so this extension is infinite. The field :\Q(\sqrt) = \left \, is an extension field of \Q, also clearly a simple extension. The degree is 2 because \left\ can serve as a basis. The field :\begin \Q\left(\sqrt, \sqrt\right) &= \Q \left(\sqrt\right) \left(\sqrt\right) \\ &= \left\ \\ &= \left\, \end is an extension field of both \Q(\sqrt) and \Q, of degree 2 and 4 respectively. It is also a simple extension, as one can show that :\begin \Q(\sqrt, \sqrt) &= \Q (\sqrt + \sqrt) \\ &= \left \. \end Finite extensions of \Q are also called algebraic number fields and are important in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
. Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of
p-adic number In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...
s \Q_p for a prime number ''p''. It is common to construct an extension field of a given field ''K'' as a
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
of the
polynomial ring 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 ...
''K'' 'X''in order to "create" a
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 su ...
for a given polynomial ''f''(''X''). Suppose for instance that ''K'' does not contain any element ''x'' with ''x''2 = −1. Then the polynomial X^2+1 is irreducible in ''K'' 'X'' consequently the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
generated by this polynomial is maximal, and L = K (X^2+1) is an extension field of ''K'' which ''does'' contain an element whose square is −1 (namely the residue class of ''X''). By iterating the above construction, one can construct a
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 ...
of any polynomial from ''K'' 'X'' This is an extension field ''L'' of ''K'' in which the given polynomial splits into a product of linear factors. If ''p'' is any
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 ...
and ''n'' is a positive integer, we have a
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 ...
GF(''pn'') with ''pn'' elements; this is an extension field of the finite field \operatorname(p) = \Z/p\Z with ''p'' elements. Given a field ''K'', we can consider the field ''K''(''X'') of all rational functions in the variable ''X'' with coefficients in ''K''; the elements of ''K''(''X'') are fractions of two
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
s over ''K'', and indeed ''K''(''X'') is the
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
of the polynomial ring ''K'' 'X'' This field of rational functions is an extension field of ''K''. This extension is infinite. Given a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
''M'', the set of all
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
s defined on ''M'' is a field, denoted by \Complex(M). It is a transcendental extension field of \Complex if we identify every complex number with the corresponding constant function defined on ''M''. More generally, given an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
''V'' over some field ''K'', then the function field of ''V'', consisting of the rational functions defined on ''V'' and denoted by ''K''(''V''), is an extension field of ''K''.


Algebraic extension

An element ''x'' of a field extension is algebraic over ''K'' if it is a
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 su ...
of a nonzero
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
with coefficients in ''K''. For example, \sqrt 2 is algebraic over the rational numbers, because it is a root of x^2-2. If an element ''x'' of ''L'' is algebraic over ''K'', the
monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\c ...
of lowest degree that has ''x'' as a root is called the minimal polynomial of ''x''. This minimal polynomial is irreducible over ''K''. An element ''s'' of ''L'' is algebraic over ''K'' if and only if the simple extension is a finite extension. In this case the degree of the extension equals the degree of the minimal polynomial, and a basis of the ''K''-
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 ...
''K''(''s'') consists of 1, s, s^2, \ldots, s^, where ''d'' is the degree of the minimal polynomial. The set of the elements of ''L'' that are algebraic over ''K'' form a subextension, which is called the
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 ( ...
of ''K'' in ''L''. This results from the preceding characterization: if ''s'' and ''t'' are algebraic, the extensions and are finite. Thus is also finite, as well as the sub extensions , and (if ). It follows that , ''st'' and 1/''s'' are all algebraic. An ''algebraic extension'' is an extension such that every element of ''L'' is algebraic over ''K''. Equivalently, an algebraic extension is an extension that is generated by algebraic elements. For example, \Q(\sqrt 2, \sqrt 3) is an algebraic extension of \Q, because \sqrt 2 and \sqrt 3 are algebraic over \Q. A simple extension is algebraic
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
it is finite. This implies that an extension is algebraic if and only if it is the union of its finite subextensions, and that every finite extension is algebraic. Every field ''K'' has an algebraic closure, which is up to an isomorphism the largest extension field of ''K'' which is algebraic over ''K'', and also the smallest extension field such that every polynomial with coefficients in ''K'' has a root in it. For example, \Complex is an algebraic closure of \R, but not an algebraic closure of \Q, as it is not algebraic over \Q (for example is not algebraic over \Q).


Transcendental extension

:''See
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 ...
for examples and more extensive discussion of transcendental extensions.'' Given a field extension , a subset ''S'' of ''L'' is called
algebraically independent In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically in ...
over ''K'' if no non-trivial polynomial relation with coefficients in ''K'' exists among the elements of ''S''. The largest cardinality of an algebraically independent set is called the
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 ...
of ''L''/''K''. It is always possible to find a set ''S'', algebraically independent over ''K'', such that ''L''/''K''(''S'') is algebraic. Such a set ''S'' is called a
transcendence basis 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 ''L''/''K''. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension. An extension ''L''/''K'' is said to be if and only if there exists a transcendence basis ''S'' of ''L''/''K'' such that ''L'' = ''K''(''S''). Such an extension has the property that all elements of ''L'' except those of ''K'' are transcendental over ''K'', but, however, there are extensions with this property which are not purely transcendental—a class of such extensions take the form ''L''/''K'' where both ''L'' and ''K'' are algebraically closed. In addition, if ''L''/''K'' is purely transcendental and ''S'' is a transcendence basis of the extension, it doesn't necessarily follow that ''L'' = ''K''(''S''). For example, consider the extension \Q(x, \sqrt)/\Q, where ''x'' is transcendental over \Q. The set \ is algebraically independent since ''x'' is transcendental. Obviously, the extension \Q(x, \sqrt)/\Q(x) is algebraic, hence \ is a transcendence basis. It doesn't generate the whole extension because there is no polynomial expression in x for \sqrt. But it is easy to see that \ is a transcendence basis that generates \Q(x, \sqrt), so this extension is indeed purely transcendental.


Normal, separable and Galois extensions

An algebraic extension ''L''/''K'' is called
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
if every
irreducible polynomial In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted f ...
in ''K'' 'X''that has a root in ''L'' completely factors into linear factors over ''L''. Every algebraic extension ''F''/''K'' admits a normal closure ''L'', which is an extension field of ''F'' such that ''L''/''K'' is normal and which is minimal with this property. An algebraic extension ''L''/''K'' is called separable if the minimal polynomial of every element of ''L'' over ''K'' is separable, i.e., has no repeated roots in an algebraic closure over ''K''. A
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 ' ...
is a field extension that is both normal and separable. A consequence of 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 extens ...
states that every finite separable extension has a primitive element (i.e. is simple). Given any field extension ''L''/''K'', we can consider its automorphism group Aut(''L''/''K''), consisting of all field automorphisms ''α'': ''L'' → ''L'' with ''α''(''x'') = ''x'' for all ''x'' in ''K''. When the extension is Galois this automorphism group is called the
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 po ...
of the extension. Extensions whose Galois group is abelian are called
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable ...
s. For a given field extension ''L''/''K'', one is often interested in the intermediate fields ''F'' (subfields of ''L'' that contain ''K''). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a bijection between the intermediate fields and the
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s of the Galois group, described by the
fundamental theorem of Galois theory In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory. In its most basi ...
.


Generalizations

Field extensions can be generalized to ring extensions which consist of a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
and one of its subrings. A closer non-commutative analog are central simple algebras (CSAs) – ring extensions over a field, which are simple algebra (no non-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field. For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are
Brauer equivalent Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik Br ...
to the reals or the quaternions. CSAs can be further generalized to
Azumaya algebra In mathematics, an Azumaya algebra is a generalization of central simple algebras to ''R''-algebras where ''R'' need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where ''R'' is a commutative local rin ...
s, where the base field is replaced by a commutative
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
.


Extension of scalars

Given a field extension, one can " extend scalars" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space via
complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
. In addition to vector spaces, one can perform extension of scalars for associative algebras defined over the field, such as polynomials or group algebras and the associated
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
s. Extension of scalars of polynomials is often used implicitly, by just considering the coefficients as being elements of a larger field, but may also be considered more formally. Extension of scalars has numerous applications, as discussed in extension of scalars: applications.


See also

* Field theory * Glossary of field theory *
Tower of fields In mathematics, a tower of fields is a sequence of field extensions : The name comes from such sequences often being written in the form :\begin\vdots \\ , \\ F_2 \\ , \\ F_1 \\ , \\ \ F_0. \end A tower of fields may be finite or infinite. Exam ...
*
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 ...
* Regular extension


Notes


References

* * * *


External links

* {{springer, title=Extension of a field, id=p/e036970