View source for Field extension

{{Use American English|date = January 2019}}
{{Short description|Construction of a larger algebraic field by "adding elements" to a smaller field}}
In [[mathematics]], particularly in [[algebra]], a '''field extension''' is a pair of [[Field (mathematics)|fields]] <math>K \subseteq L</math>, such that the operations of ''K'' are those of ''L'' [[Restriction (mathematics)|restricted]] to ''K''. In this case, ''L'' is an '''extension field''' of ''K'' and ''K'' is a '''subfield''' of ''L''.<ref>{{harvtxt|Fraleigh|1976|p=293}}</ref><ref>{{harvtxt|Herstein|1964|p=167}}</ref><ref>{{harvtxt|McCoy|1968|p=116}}</ref> For example, under the usual notions of [[addition]] and [[multiplication]], the [[complex number]]s are an extension field of the [[real number]]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]], and are widely used in [[algebraic geometry]].

==Subfield==
A '''subfield''' <math>K</math> of a [[field (mathematics)|field]] <math>L</math> is a [[subset]] <math>K\subseteq L</math> that is a field with respect to the field operations inherited from <math>L</math>. Equivalently, a subfield is a subset that contains the [[multiplicative identity]] <math>1</math>, and is [[Closure (mathematics)|closed]] under the operations of addition, subtraction, multiplication, and taking the [[multiplicative inverse|inverse]] of a nonzero element of <math>K</math>.

As {{math|1=1 – 1 = 0}}, the latter definition implies <math>K</math> and <math>L</math> have the same [[zero element]].

For example, the field of [[rational number]]s is a subfield of the [[real number]]s, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is [[isomorphism|isomorphic]] to) a subfield of any field of [[characteristic of a ring|characteristic]] <math>0</math>.

The [[characteristic (algebra)|characteristic]] of a subfield is the same as the characteristic of the larger field.

==Extension field==

If <math>K</math> is a subfield of <math>L</math>, then <math>L</math> is an '''extension field''' or simply '''extension''' of <math>K</math>, and this pair of fields is a '''field extension'''. Such a field extension is denoted <math>L/K</math> (read as "<math>L</math> over <math>K</math>").

If <math>L</math> is an extension of <math>F</math>, which is in turn an extension of <math>K</math>, then <math>F</math> is said to be an '''intermediate field''' (or '''intermediate extension''' or '''subextension''') of <math>L/K</math>.

Given a field extension <math>L/K</math>, the larger field <math>L</math> is a <math>K</math>-[[vector space]]. The [[dimension (vector space)|dimension]] of this vector space is called the [[degree of a field extension|'''degree''' of the extension]] and is denoted by <math>[L:K]</math>. 

The degree of an extension is 1 if and only if the two fields are equal. In this case, the extension is a '''{{vanchor|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 <math>L/K</math> and <math>M/L</math>, the extension <math>M/K</math> is finite if and only if both <math>L/K</math> and <math>M/L</math> are finite. In this case, one has

:<math>[M : K]=[M : L]\cdot[L : K].</math>

Given a field extension <math>L/K</math> and a subset <math>S</math> of <math>L</math>, there is a smallest subfield of <math>L</math> that contains <math>K</math> and <math>S</math>. It is the intersection of all subfields of <math>L</math> that contain <math>K</math> and <math>S</math>, and is denoted by <math>K(S)</math> (read as "<math>K</math> ''{{vanchor|adjoin}}'' <math>S</math>"). One says that <math>K(S)</math> is the field ''generated'' by <math>S</math> over <math>K</math>, and that <math>S</math> is a [[generating set]] of <math>K(S)</math> over <math>K</math>. When <math>S=\{x_1, \ldots, x_n\}</math> is finite, one writes <math>K(x_1, \ldots, x_n)</math> instead of <math>K(\{x_1, \ldots, x_n\}),</math> and one says that <math>K(S)</math> is {{vanchor|finitely generated}} over <math>K</math>. If <math>S</math> consists of a single element <math>s</math>, the extension <math>K(s)/K</math> is called a [[simple extension]]<ref>{{harvtxt|Fraleigh|1976|p=298}}</ref><ref>{{harvtxt|Herstein|1964|p=193}}</ref> and <math>s</math> is called a [[primitive element (field theory)|primitive element]] of the extension.<ref>{{harvtxt|Fraleigh|1976|p=363}}</ref>

An extension field of the form <math>K(S)</math> is often said to result from the ''{{vanchor|adjunction}}'' of <math>S</math> to <math>K</math>.<ref>{{harvtxt|Fraleigh|1976|p=319}}</ref><ref>{{harvtxt|Herstein|1964|p=169}}</ref>

In [[characteristic of a ring|characteristic]] 0, every finite extension is a simple extension. This is the [[primitive element theorem]], which does not hold true for fields of non-zero characteristic.

If a simple extension <math>K(s)/K</math> is not finite, the field <math>K(s)</math> is isomorphic to the field of [[rational fraction]]s in <math>s</math> over <math>K</math>.

== Caveats ==
The notation ''L'' / ''K'' is purely formal and does not imply the formation of a [[quotient ring]] 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 function|injective]] [[ring homomorphism]] between two fields.
''Every'' non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper [[Ideal_(ring_theory)|ideals]], so field extensions are precisely the [[morphism]]s in the [[category of fields]].

Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.

== Examples ==
The field of complex numbers <math>\Complex</math> is an extension field of the field of [[real number]]s <math>\R</math>, and <math>\R</math> in turn is an extension field of the field of rational numbers <math>\Q</math>. Clearly then, <math>\Complex/\Q</math> is also a field extension. We have <math>[\Complex:\R] =2</math> because <math>\{1, i\}</math> is a basis, so the extension <math>\Complex/\R</math> is finite. This is a simple extension because <math>\Complex = \R(i).</math> <math>[\R:\Q] =\mathfrak c</math> (the [[cardinality of the continuum]]), so this extension is infinite.

The field 

:<math>\Q(\sqrt{2}) =  \left \{ a + b\sqrt{2} \mid a,b \in \Q \right \},</math>

is an extension field of <math>\Q,</math> also clearly a simple extension. The degree is 2 because <math>\left\{1, \sqrt{2}\right\}</math> can serve as a basis. 

The field

:<math>\begin{align}
\Q\left(\sqrt{2}, \sqrt{3}\right) &= \Q \left(\sqrt{2}\right) \left(\sqrt{3}\right) \\
&= \left\{ a+b\sqrt{3} \mid a,b \in \Q\left(\sqrt{2}\right) \right\} \\
&= \left\{ a + b \sqrt{2} + c\sqrt{3} + d\sqrt{6} \mid a,b,c, d \in \Q \right\},
\end{align}</math>

is an extension field of both <math>\Q(\sqrt{2})</math> and <math>\Q,</math> of degree 2 and 4 respectively. It is also a simple extension, as one can show that 

:<math>\begin{align}
\Q(\sqrt{2}, \sqrt{3}) &= \Q (\sqrt{2} + \sqrt{3}) \\
&= \left \{ a + b (\sqrt{2} + \sqrt{3}) + c (\sqrt{2} + \sqrt{3})^2 + d(\sqrt{2} + \sqrt{3})^3 \mid a,b,c, d \in \Q\right\}.
\end{align}</math>

Finite extensions of <math>\Q</math> are also called [[algebraic number field]]s and are important in [[number theory]]. 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]]s <math>\Q_p</math> for a prime number ''p''.

It is common to construct an extension field of a given field ''K'' as a [[quotient ring]] of the [[polynomial ring]] ''K''[''X''] in order to "create" a [[root of a function|root]] for a given polynomial ''f''(''X''). Suppose for instance that ''K'' does not contain any element ''x'' with ''x''<sup>2</sup> = −1. Then the polynomial <math>X^2+1</math> is [[irreducible polynomial|irreducible]] in ''K''[''X''], consequently the [[ideal (ring theory)|ideal]] generated by this polynomial is [[maximal ideal|maximal]], and <math>L = K[X]/(X^2+1)</math> is an extension field of ''K'' which ''does'' contain an element whose square is −1 (namely the [[modular arithmetic|residue class]] of ''X'').

By iterating the above construction, one can construct a [[splitting field]] 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]] and ''n'' is a positive integer, there is a unique (up to isomorphism) [[finite field]] <math>GF(p^n) = \mathbb{F}_{p^n}</math> with ''p<sup>n</sup>'' elements; this is an extension field of the [[prime field]] <math>\operatorname{GF}(p) = \mathbb{F}_p = \Z/p\Z</math> with ''p'' elements.

Given a field ''K'', we can consider the field ''K''(''X'') of all [[rational function]]s in the variable ''X'' with coefficients in ''K''; the elements of ''K''(''X'') are fractions of two [[polynomial]]s over ''K'', and indeed ''K''(''X'') is the [[field of fractions]] 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]] ''M'', the set of all [[meromorphic function]]s defined on ''M'' is a field, denoted by <math>\Complex(M).</math> It is a transcendental extension field of <math>\Complex</math> if we identify every complex number with the corresponding [[constant function]] defined on ''M''. More generally, given an [[algebraic variety]] ''V'' over some field ''K'', the [[function field of an algebraic variety|function field]] ''K''(''V''), consisting of the rational functions defined on ''V'', is an extension field of ''K''.

== Algebraic extension ==
{{main|Algebraic extension|Algebraic element}}
An element ''x'' of a field extension <math>L/K</math> is algebraic over ''K'' if it is a [[root of a function|root]] of a nonzero [[polynomial]] with coefficients in ''K''. For example, <math>\sqrt 2</math> is algebraic over the rational numbers, because it is a root of <math>x^2-2.</math> If an element ''x'' of ''L'' is algebraic over ''K'', the [[monic polynomial]] of lowest degree that has ''x'' as a root is called the [[minimal polynomial (field theory)|minimal polynomial]] of ''x''. This minimal polynomial is [[irreducible polynomial|irreducible]] over ''K''.

An element ''s'' of ''L'' is algebraic over ''K'' if and only if the simple extension {{nowrap|''K''(''s'') /''K''}} 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]] ''K''(''s'') consists of <math>1, s, s^2, \ldots, s^{d-1},</math> 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]] of ''K'' in ''L''. This results from the preceding characterization: if ''s'' and ''t'' are algebraic, the extensions {{nowrap|''K''(''s'') /''K''}} and {{nowrap|''K''(''s'')(''t'') /''K''(''s'')}} are finite. Thus {{nowrap|''K''(''s'', ''t'') /''K''}} is also finite, as well as the sub extensions {{nowrap|''K''(''s'' ± ''t'') /''K''}}, {{nowrap|''K''(''st'') /''K''}} and {{nowrap|''K''(1/''s'') /''K''}} (if {{nowrap|''s'' ≠ 0}}). It follows that {{nowrap|''s'' ± ''t''}}, ''st'' and 1/''s'' are all algebraic.

An ''algebraic extension'' <math>L/K</math> 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, <math>\Q(\sqrt 2, \sqrt 3)</math> is an algebraic extension of <math>\Q</math>, because <math>\sqrt 2</math> and <math>\sqrt 3</math> are algebraic over <math>\Q.</math>

A simple extension is algebraic [[if and only if]] 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, <math>\Complex</math> is an algebraic closure of <math>\R</math>, but not an algebraic closure of <math>\Q</math>, as it is not algebraic over <math>\Q</math> (for example {{pi}} is not algebraic over <math>\Q</math>).

==Transcendental extension==
{{main|Transcendental extension}}
Given a field extension <math>L/K</math>, a subset ''S'' of ''L'' is called [[algebraically independent]] 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]] 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]] of ''L''/''K''. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension. An extension <math>L/K</math> is said to be '''{{visible anchor|purely transcendental}}''' if and only if there exists a transcendence basis ''S'' of <math>L/K</math> 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. 

If ''L''/''K'' is purely transcendental and ''S'' is a transcendence basis of the extension, it doesn't necessarily follow that ''L'' = ''K''(''S''). On the opposite, even when one knows a transcendence basis, it may be difficult to decide whether the extension is purely separable, and if it is so, it may be difficult to find a transcendence basis ''S'' such that ''L'' = ''K''(''S'').

For example, consider the extension <math>\Q(x, y)/\Q,</math> where <math>x</math> is transcendental over <math>\Q,</math> and <math>y</math> is a [[polynomial root|root]] of the equation <math>y^2-x^3=0.</math> Such an extension can be defined as <math>\Q(X)[Y]/\langle Y^2-X^3\rangle,</math> in which <math>x</math> and <math>y</math> are the [[equivalence class]]es of <math>X</math> and <math>Y.</math> Obviously, the singleton set <math>\{x\}</math> is transcendental over <math>\Q</math> and the extension <math>\Q(x, y)/\Q(x)</math> is algebraic; hence <math>\{x\}</math> is a transcendence basis that does not generates the extension <math>\Q(x, y)/\Q(x)</math>. Similarly, <math>\{y\}</math> is a transcendence basis that does not generates the whole extension. However the extension is purely transcendental since, if one set <math>t=y/x,</math> one has <math>x=t^2</math> and <math>y=t^3,</math> and thus <math>t</math> generates the whole extension. 

Purely transcendental extensions of an algebraically closed field occur as [[function field of an algebraic variety|function fields]] of [[rational varieties]]. The problem of finding a [[rational parametrization]] of a rational variety is equivalent with the problem of finding a transcendence basis that generates the whole extension.

== Normal, separable and Galois extensions ==
An algebraic extension <math>L/K</math> is called [[normal extension|normal]] if every [[irreducible polynomial]] 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 <math>L/K</math> is normal and which is minimal with this property.

An algebraic extension <math>L/K</math> is called [[separable extension|separable]] if the minimal polynomial of every element of ''L'' over ''K'' is [[separable polynomial|separable]], i.e., has no repeated roots in an algebraic closure over ''K''. A [[Galois extension]] is a field extension that is both normal and separable.

A consequence of the [[primitive element theorem]] states that every finite separable extension has a primitive element (i.e. is simple).

Given any field extension <math>L/K</math>, we can consider its '''automorphism group''' <math>\text{Aut}(L/K)</math>, consisting of all field [[automorphism]]s ''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]] of the extension. Extensions whose Galois group is [[abelian group|abelian]] are called [[abelian extension]]s.

For a given field extension <math>L/K</math>, 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]]s of the Galois group, described by the [[fundamental theorem of Galois theory]].

== Generalizations ==
Field extensions can be generalized to [[ring extensions]] which consist of a [[ring (mathematics)|ring]] and one of its [[subring]]s. A closer non-commutative analog are [[central simple algebra]]s (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_(ring_theory)|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]] to the reals or the quaternions. CSAs can be further generalized to [[Azumaya algebra]]s, where the base field is replaced by a commutative [[local ring]].

== Extension of scalars ==
{{main|Extension of scalars}}
Given a field extension, one can "[[Extension of scalars|extend scalars]]" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space via [[complexification]]. In addition to vector spaces, one can perform extension of scalars for [[associative algebra]]s defined over the field, such as polynomials or [[group ring|group algebra]]s and the associated [[group representation]]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|extension of scalars: applications]].

==See also==
{{wiktionary|field extension}}
{{wiktionary|extension field}}

* [[Field theory (mathematics)|Field theory]]
* [[Glossary of field theory]]
* [[Tower of fields]]
* [[Primary extension]]
* [[Regular extension]]

== Notes ==
<references/>

==References==
*{{citation|first1 = John B. | last1 = Fraleigh | year = 1976 | isbn = 0-201-01984-1 | title = A First Course In Abstract Algebra | edition = 2nd | publisher = [[Addison-Wesley]] | location = Reading }}
*{{citation|first1 = I. N. | last1 = Herstein | year = 1964 | isbn = 978-1114541016 | title = Topics In Algebra | publisher = [[Blaisdell Publishing Company]] | location = Waltham }}
*{{Lang Algebra | edition=3r2004}}
*{{citation|first1 = Neal H. | last1 = McCoy | year = 1968 | title = Introduction To Modern Algebra, Revised Edition | publisher = [[Allyn and Bacon]] | location = Boston | lccn = 68015225 }}

==External links==
* {{springer|title=Extension of a field|id=p/e036970}}

[[Category:Field extensions| ]]