Field extension: Difference between revisions

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{{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.

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]].

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== 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.~~

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". Some authors use the notations ''L'' : ''K'' or ''L'' | ''K'', while others may simply indicate verbally that <math>L\supset K

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'' 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]].

</math> is a field extension. Towers of extensions are often depicted diagrammatically. For example, the diagram below depicts the situation where ''L'' is an extension of ''K'' and ''K'' is an extension of ''F'':

:<math>\begin{array}{c} L \\ \Big| \\ K \\ \Big| \\ F \end{array}

</math>

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'' 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]]{{Clarify|date=April 2026|reason=this sentence could be partially wrong, see talk page}}.

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

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==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.

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 generate the extension <math>\Q(x, y)/\Q(x)</math>. Similarly, <math>\{y\}</math> is a transcendence basis that does not generate 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.

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== 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]].

@@ Line 1: / Line 1: @@
 {{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.
+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]].
@@ Line 37: / Line 37: @@
 == 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.
+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". Some authors use the notations ''L'' : ''K'' or ''L'' | ''K'', while others may simply indicate verbally that <math>L\supset K
-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'' 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]].
+</math> is a field extension.  Towers of extensions are often depicted diagrammatically.  For example, the diagram below depicts the situation where ''L'' is an extension of ''K'' and ''K'' is an extension of ''F'':
+:<math>\begin{array}{c} L \\ \Big| \\ K \\ \Big| \\ F \end{array}
+</math>
+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'' 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]]{{Clarify|date=April 2026|reason=this sentence could be partially wrong, see talk page}}.
 Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.
@@ Line 96: / Line 103: @@
 ==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.
+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 generate the extension <math>\Q(x, y)/\Q(x)</math>. Similarly, <math>\{y\}</math> is a transcendence basis that does not generate 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.
@@ Line 120: / Line 128: @@
 == 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]].

Field extension: Difference between revisions

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