Editing Algebraic closure

{{Short description|Algebraic field extension}}
{{About||the set of end points of line segments contained in a subset of a vector space|Algebraic closure (convex analysis)}}

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|
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''.

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 extension|algebraic over]] ''K'' form an algebraic closure of ''K''.

The algebraic closure of a field ''K'' has the same [[cardinal number|cardinality]] as ''K'' if ''K'' is infinite, and is [[countably infinite]] if ''K'' is finite.<ref name=Kap7476/>

==Examples==
*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.
*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).
*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>

==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''].
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
: <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>.
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'',
Zorn's lemma implies that there exists a maximal ideal ''M'' in ''R'' that contains ''I''.
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. 

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''.

==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>

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.

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>

== See also==
* [[Algebraically closed field]]
* [[Algebraic extension]]
* [[Puiseux expansion]]
*[[Complete field]]

==References==
{{reflist}}
* {{cite book | first=Irving | last=Kaplansky | author-link=Irving Kaplansky | title=Fields and rings | edition=Second | series=Chicago lectures in mathematics | publisher=University of Chicago Press | year=1972 | isbn=0-226-42451-0 | zbl=1001.16500 }}
* {{cite book | last=McCarthy | first=Paul J. | title=Algebraic extensions of fields | edition=Corrected reprint of the 2nd | zbl=0768.12001 | location=New York | publisher=Dover Publications | year=1991 }}

{{DEFAULTSORT:Algebraic Closure}}
[[Category:Field extensions]]