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In [[mathematics]], particularly [[abstract algebra]], an '''algebraic closure''' of a [[field (mathematics)|field]] ''K'' is an [[algebraic extension]] of ''K'' that is [[algebraically closed field|algebraically closed]].  It is one of many [[closure (mathematics)|closures]] in mathematics.
In [[mathematics]], particularly [[abstract algebra]], an '''algebraic closure''' of a [[field (mathematics)|field]] ''K'' is an [[algebraic extension]] of ''K'' that is [[algebraically closed field|algebraically closed]].  It is one of many [[closure (mathematics)|closures]] in mathematics.


Using [[Zorn's lemma]]<ref name=McC21>McCarthy (1991) p.21</ref><ref>[[Michael Atiyah|M. F. Atiyah]] and [[I. G. Macdonald]] (1969). ''Introduction to commutative algebra''. Addison-Wesley publishing Company. pp.&nbsp;11–12.</ref><ref name=Kap7476>Kaplansky (1972) pp.74-76</ref> or the weaker [[ultrafilter lemma]],<ref>{{Citation|first=Bernhard|last=Banaschewski|
Using [[Zorn's lemma]]<ref name=McC21>McCarthy (1991) p.21</ref><ref>[[Michael Atiyah|M. F. Atiyah]] and [[I. G. Macdonald]] (1969). ''[[Introduction to Commutative Algebra]]''. Addison-Wesley publishing Company. pp.&nbsp;11–12.</ref><ref name=Kap7476>Kaplansky (1972) pp.74-76</ref> or the weaker [[ultrafilter lemma]],<ref>{{Citation|first=Bernhard|last=Banaschewski|
title=Algebraic closure without choice.|journal=Z. Math. Logik Grundlagen Math.|volume=38|issue=4|pages=383–385|year=1992|doi=10.1002/malq.19920380136|zbl=0739.03027}}</ref><ref>[https://mathoverflow.net/questions/46566/is-the-statement-that-every-field-has-an-algebraic-closure-known-to-be-equivalent Mathoverflow discussion]</ref>  it can be shown that [[#Existence of an algebraic closure and splitting fields|every field has an algebraic closure]], and that the algebraic closure of a field ''K'' is unique [[up to]] an [[isomorphism]] that [[fixed point (mathematics)|fixes]] every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''.
title=Algebraic closure without choice.|journal=Z. Math. Logik Grundlagen Math.|volume=38|issue=4|pages=383–385|year=1992|doi=10.1002/malq.19920380136|zbl=0739.03027}}</ref><ref>[https://mathoverflow.net/questions/46566/is-the-statement-that-every-field-has-an-algebraic-closure-known-to-be-equivalent Mathoverflow discussion]</ref>  it can be shown that [[#Existence of an algebraic closure and splitting fields|every field has an algebraic closure]], and that the algebraic closure of a field ''K'' is unique [[up to]] an [[isomorphism]] that [[fixed point (mathematics)|fixes]] every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''.


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*The [[fundamental theorem of algebra]] states that the algebraic closure of the field of [[real number]]s is the field of [[complex number]]s.
*The [[fundamental theorem of algebra]] states that the algebraic closure of the field of [[real number]]s is the field of [[complex number]]s.
*The algebraic closure of the field of [[rational number]]s is the field of [[algebraic number]]s.
*The algebraic closure of the field of [[rational number]]s is the field of [[algebraic number]]s.
*There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of [[transcendental extension]]s of the rational numbers, e.g. the algebraic closure of '''Q'''(π).
*There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of [[transcendental extension]]s of the rational numbers, e.g. the algebraic closure of <math>\mathbf{Q} (\pi)</math>.
*For a [[finite field]] of [[prime number|prime]] power order ''q'', the algebraic closure is a [[countably infinite]] field that contains a copy of the field of order ''q''<sup>''n''</sup> for each positive [[integer]] ''n'' (and is in fact the union of these copies).<ref>{{citation | title=Infinite Algebraic Extensions of Finite Fields | volume=95 | series=Contemporary Mathematics | first1=Joel V. | last1=Brawley | first2=George E. | last2=Schnibben | publisher=[[American Mathematical Society]] | year=1989 | isbn=978-0-8218-5428-0 | contribution=2.2 The Algebraic Closure of a Finite Field | pages=22–23 | url=https://books.google.com/books?id=0HNfpAsMXhUC&pg=PA22 | zbl=0674.12009}}.</ref>
*For a [[finite field]] of [[prime number|prime]] power order ''<math>q</math>'', the algebraic closure is a [[countably infinite]] field that contains a copy of the field of order ''<math>q^n</math>'' for each positive [[integer]] ''<math>n</math>'' (and is in fact the union of these copies).<ref>{{citation | title=Infinite Algebraic Extensions of Finite Fields | volume=95 | series=Contemporary Mathematics | first1=Joel V. | last1=Brawley | first2=George E. | last2=Schnibben | publisher=[[American Mathematical Society]] | year=1989 | isbn=978-0-8218-5428-0 | contribution=2.2 The Algebraic Closure of a Finite Field | pages=22–23 | url=https://books.google.com/books?id=0HNfpAsMXhUC&pg=PA22 | zbl=0674.12009}}.</ref>


==Existence of an algebraic closure and splitting fields==
==Existence of an algebraic closure and splitting fields==
Let <math>S = \{ f_{\lambda} | \lambda \in \Lambda\}</math> be the set of all [[monic polynomial|monic]] [[irreducible polynomial]]s in ''K''[''x''].
Let <math>S = \{ f_{\lambda} \mid \lambda \in \Lambda\}</math> be the set of all [[monic polynomial|monic]] [[irreducible polynomial]]s in <math>K[x]</math>.
For each <math>f_{\lambda} \in S</math>, introduce new variables <math>u_{\lambda,1},\ldots,u_{\lambda,d}</math> where <math>d = {\rm degree}(f_{\lambda})</math>.
For each <math>f_{\lambda} \in S</math>, introduce new variables <math>u_{\lambda,1},\ldots,u_{\lambda,d}</math> where <math>d = {\rm degree}(f_{\lambda})</math>.
Let ''R'' be the polynomial ring over ''K'' generated by <math>u_{\lambda,i}</math> for all <math>\lambda \in \Lambda</math> and all <math>i \leq {\rm degree}(f_{\lambda}).</math> Write
Let <math>R</math> be the polynomial ring over <math>K</math> generated by <math>u_{\lambda,i}</math> for all <math>\lambda \in \Lambda</math> and all <math>i \leq {\rm degree}(f_{\lambda}).</math> Write
: <math>f_{\lambda} - \prod_{i=1}^d (x-u_{\lambda,i}) = \sum_{j=0}^{d-1} r_{\lambda,j} \cdot x^j \in R[x]</math>
: <math>f_{\lambda} - \prod_{i=1}^d (x-u_{\lambda,i}) = \sum_{j=0}^{d-1} r_{\lambda,j} \cdot x^j \in R[x]</math>


with <math>r_{\lambda,j} \in R</math>.
with <math>r_{\lambda,j} \in R</math>.
Let  ''I'' be the [[ideal of a ring|ideal]] in ''R'' generated by the <math>r_{\lambda,j}</math>. Since ''I'' is strictly smaller than  ''R'',
Let  <math>I</math> be the [[ideal of a ring|ideal]] in <math>R</math> generated by the <math>r_{\lambda,j}</math>. Since <math>I</math> is strictly smaller than  <math>R</math>,
Zorn's lemma implies that there exists a maximal ideal ''M'' in ''R'' that contains ''I''.
Zorn's lemma implies that there exists a maximal ideal <math>M</math> in <math>R</math> that contains <math>I</math>.
The field ''K''<sub>1</sub>=''R''/''M'' has the property that every polynomial <math>f_{\lambda}</math> with coefficients in ''K'' splits as the product of <math>x-(u_{\lambda,i} + M),</math> and hence has all roots in ''K''<sub>1</sub>. In the same way, an extension ''K''<sub>2</sub> of ''K''<sub>1</sub> can be constructed, etc. The union of all these extensions is the algebraic closure of ''K'', because any polynomial with coefficients in this new field has its coefficients in some ''K''<sub>''n''</sub> with sufficiently large ''n'', and then its roots are in ''K''<sub>''n''+1</sub>, and hence in the union itself.  
The field <math>K_1=R/M</math> has the property that every polynomial <math>f_{\lambda}</math> with coefficients in <math>K</math> splits as the product of <math>x-(u_{\lambda,i} + M),</math> and hence has all roots in <math>K_1</math>. In the same way, an extension <math>K_2</math> of <math>K_1</math> can be constructed, etc. The union of all these extensions is the algebraic closure of <math>K</math>, because any polynomial with coefficients in this new field has its coefficients in some <math>K_n</math> with sufficiently large <math>n</math>, and then its roots are in <math>K_{n+1}</math>, and hence in the union itself.  


It can be shown along the same lines that for any subset ''S'' of ''K''[''x''], there exists a [[splitting field]] of ''S'' over ''K''.
It can be shown along the same lines that for any subset <math>S</math> of <math>K[x]</math>, there exists a [[splitting field]] of <math>S</math> over <math>K</math>.


==Separable closure==
==Separable closure==
An algebraic closure ''K<sup>alg</sup>'' of ''K'' contains a unique [[separable extension]] ''K<sup>sep</sup>'' of ''K'' containing all (algebraic) separable extensions of ''K'' within ''K<sup>alg</sup>''.  This subextension is called a '''separable closure''' of ''K''. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of ''K<sup>sep</sup>'', of degree > 1. Saying this another way, ''K'' is contained in a ''separably-closed'' algebraic extension field. It is unique ([[up to]] isomorphism).<ref name=McC22>McCarthy (1991) p.22</ref>
An algebraic closure <math>K^{\text{alg}}</math>of ''<math>K</math>'' contains a unique [[separable extension]] ''<math>K^{\text{sep}}</math>'' of ''K'' containing all (algebraic) separable extensions of ''<math>K</math>'' within <math>K^{\text{alg}}</math>.  This subextension is called a '''separable closure''' of ''<math>K</math>''. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of <math>K^{\text{sep}}</math>, of degree > 1. Saying this another way, ''<math>K</math>'' is contained in a ''separably-closed'' algebraic extension field. It is unique ([[up to]] isomorphism).<ref name=McC22>McCarthy (1991) p.22</ref>


The separable closure is the full algebraic closure if and only if ''K'' is a [[perfect field]]. For example, if ''K'' is a field of [[characteristic of a field|characteristic]] ''p'' and if ''X'' is transcendental over ''K'', <math>K(X)(\sqrt[p]{X}) \supset K(X)</math> is a non-separable algebraic field extension.
The separable closure is the full algebraic closure if and only if ''<math>K</math>'' is a [[perfect field]]. For example, if ''<math>K</math>'' is a field of [[characteristic of a field|characteristic]] ''<math>p</math>'' and if ''<math>X</math>'' is transcendental over ''<math>K</math>'', <math>K(X)(\sqrt[p]{X}) \supset K(X)</math> is a non-separable algebraic field extension.


In general, the [[absolute Galois group]] of ''K'' is the Galois group of ''K<sup>sep</sup>'' over ''K''.<ref name=FJ12>{{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=[[Springer-Verlag]] | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 | page=12 }}</ref>
In general, the [[absolute Galois group]] of ''<math>K</math>'' is the Galois group of <math>K^{\text{sep}}</math> over ''<math>K</math>''.<ref name=FJ12>{{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe |author-link=Michael D. Fried |author-link2=Moshe Jarden | title=Field arithmetic | edition=3rd | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=[[Springer-Verlag]] | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 | page=12 }}</ref>


== See also==
== See also==

Latest revision as of 07:09, 17 May 2026

For the set of end points of line segments contained in a subset of a vector space, see Algebraic closure (convex analysis).

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 lemma[1][2][3] or the weaker ultrafilter lemma,[4][5] it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

The algebraic closure of a field K can be thought of as the largest algebraic extension of K. To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed field containing K, because if M is any algebraically closed field containing K, then the elements of M that are algebraic over K form an algebraic closure of K.

The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.[3]

Examples

  • The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.
  • The algebraic closure of the field of rational numbers is the field of algebraic numbers.
  • There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{Q} (\pi)} .
  • For a finite field of prime power order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} , the algebraic closure is a countably infinite field that contains a copy of the field of order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^n} for each positive integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} (and is in fact the union of these copies).[6]

Existence of an algebraic closure and splitting fields

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S = \{ f_{\lambda} \mid \lambda \in \Lambda\}} be the set of all monic irreducible polynomials in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K[x]} . For each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{\lambda} \in S} , introduce new variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{\lambda,1},\ldots,u_{\lambda,d}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = {\rm degree}(f_{\lambda})} . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} be the polynomial ring over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} generated by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_{\lambda,i}} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda \in \Lambda} and all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \leq {\rm degree}(f_{\lambda}).} Write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{\lambda} - \prod_{i=1}^d (x-u_{\lambda,i}) = \sum_{j=0}^{d-1} r_{\lambda,j} \cdot x^j \in R[x]}

with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{\lambda,j} \in R} . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} be the ideal in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} generated by the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{\lambda,j}} . Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is strictly smaller than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , Zorn's lemma implies that there exists a maximal ideal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} that contains Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} . The field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_1=R/M} has the property that every polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{\lambda}} with coefficients in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} splits as the product of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x-(u_{\lambda,i} + M),} and hence has all roots in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_1} . In the same way, an extension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_2} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_1} can be constructed, etc. The union of all these extensions is the algebraic closure of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} , because any polynomial with coefficients in this new field has its coefficients in some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_n} with sufficiently large Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , and then its roots are in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_{n+1}} , and hence in the union itself.

It can be shown along the same lines that for any subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K[x]} , there exists a splitting field of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} .

Separable closure

An algebraic closure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^{\text{alg}}} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} contains a unique separable extension Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^{\text{sep}}} of K containing all (algebraic) separable extensions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} within Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^{\text{alg}}} . This subextension is called a separable closure of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} . Since a separable extension of a separable extension is again separable, there are no finite separable extensions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^{\text{sep}}} , of degree > 1. Saying this another way, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is contained in a separably-closed algebraic extension field. It is unique (up to isomorphism).[7]

The separable closure is the full algebraic closure if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is a perfect field. For example, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is a field of characteristic Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} and if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is transcendental over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K(X)(\sqrt[p]{X}) \supset K(X)} is a non-separable algebraic field extension.

In general, the absolute Galois group of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is the Galois group of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^{\text{sep}}} over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} .[8]

See also

References

  1. ↑ McCarthy (1991) p.21
  2. ↑ M. F. Atiyah and I. G. Macdonald (1969). Introduction to Commutative Algebra. Addison-Wesley publishing Company. pp. 11–12.
  3. ↑ 3.0 3.1 Kaplansky (1972) pp.74-76
  4. ↑ Banaschewski, Bernhard (1992), "Algebraic closure without choice.", Z. Math. Logik Grundlagen Math., 38 (4): 383–385, doi:10.1002/malq.19920380136, Zbl 0739.03027
  5. ↑ Mathoverflow discussion
  6. ↑ Brawley, Joel V.; Schnibben, George E. (1989), "2.2 The Algebraic Closure of a Finite Field", Infinite Algebraic Extensions of Finite Fields, Contemporary Mathematics, 95, American Mathematical Society, pp. 22–23, ISBN 978-0-8218-5428-0, Zbl 0674.12009.
  7. ↑ McCarthy (1991) p.22
  8. ↑ Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 11 (3rd ed.). Springer-Verlag. p. 12. ISBN 978-3-540-77269-9. Zbl 1145.12001.
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