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In [[field theory (mathematics)|field theory]], a branch of algebra, a [[field extension]] <math>L/k</math> is said to be '''regular''' if ''k'' is [[algebraically closed]] in ''L''  {{Clarification needed|date=July 2013}} and ''L'' is [[separable extension|separable]] over ''k'', or equivalently, <math>L \otimes_k \overline{k}</math> is an integral domain when <math>\overline{k}</math> is the algebraic closure of <math>k</math> (that is, to say, <math>L, \overline{k}</math> are [[linearly disjoint]] over ''k'').<ref name=FJ38>Fried & Jarden (2008) p.38</ref><ref name=C425>Cohn (2003) p.425</ref>


==Properties==
* Regularity is transitive: if ''F''/''E'' and ''E''/''K'' are regular then so is ''F''/''K''.<ref name=FJ39>Fried & Jarden (2008) p.39</ref>
* If ''F''/''K'' is regular then so is ''E''/''K'' for any ''E'' between ''F'' and ''K''.<ref name=FJ39/>
* The extension ''L''/''k'' is regular if and only if every subfield of ''L'' finitely generated over ''k'' is regular over ''k''.<ref name=C425/>
* Any extension of an algebraically closed field is regular.<ref name=FJ39/><ref name=C426>Cohn (2003) p.426</ref>
* An extension is regular if and only if it is separable and [[primary extension|primary]].<ref name=FJ44>Fried & Jarden (2008) p.44</ref>
* A [[purely transcendental extension]] of a field is regular.


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==Self-regular extension==
There is also a similar notion: a field extension <math>L / k</math> is said to be '''self-regular''' if <math>L \otimes_k L</math> is an integral domain. A self-regular extension is relatively algebraically closed in ''k''.<ref name=C427>Cohn (2003) p.427</ref> However, a self-regular extension is not necessarily regular.{{Citation needed|date=February 2010}}
 
==References==
{{reflist}}
* {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd revised | 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 | pages=38-41 }}
* M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese) [http://www.shokabo.co.jp/mybooks/ISBN978-4-7853-1309-8.htm]
* {{cite book | title=Basic Algebra. Groups, Rings, and Fields | first=P. M. | last=Cohn | authorlink=Paul Cohn | publisher=[[Springer-Verlag]] | year=2003 | isbn=1-85233-587-4 | zbl=1003.00001 }}
* A. Weil, [[Foundations of algebraic geometry]].
 
[[Category:Field theory]]
 
 
{{Abstract-algebra-stub}}

Latest revision as of 15:37, 12 December 2013

In field theory, a branch of algebra, a field extension is said to be regular if k is algebraically closed in L Template:Clarification needed and L is separable over k, or equivalently, is an integral domain when is the algebraic closure of (that is, to say, are linearly disjoint over k).[1][2]

Properties

  • Regularity is transitive: if F/E and E/K are regular then so is F/K.[3]
  • If F/K is regular then so is E/K for any E between F and K.[3]
  • The extension L/k is regular if and only if every subfield of L finitely generated over k is regular over k.[2]
  • Any extension of an algebraically closed field is regular.[3][4]
  • An extension is regular if and only if it is separable and primary.[5]
  • A purely transcendental extension of a field is regular.

Self-regular extension

There is also a similar notion: a field extension is said to be self-regular if is an integral domain. A self-regular extension is relatively algebraically closed in k.[6] However, a self-regular extension is not necessarily regular.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

References

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Template:Abstract-algebra-stub

  1. ↑ Fried & Jarden (2008) p.38
  2. ↑ 2.0 2.1 Cohn (2003) p.425
  3. ↑ 3.0 3.1 3.2 Fried & Jarden (2008) p.39
  4. ↑ Cohn (2003) p.426
  5. ↑ Fried & Jarden (2008) p.44
  6. ↑ Cohn (2003) p.427
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