{{Short description|Type of field extension}} In 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'' (i.e., <math>k = \hat k</math> where <math>\hat k</math> is the set of elements in ''L'' algebraic over ''k'') and ''L'' is 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.<ref name=FJ44>Fried & Jarden (2008) p.44</ref> * A purely transcendental extension of a field is regular.
==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 |author-link=Michael D. Fried |author-link2=Moshe Jarden | 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 extensions
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