Regular extension

In field theory, a branch of algebra, a field extension $L/k$ is said to be regular if k is algebraically closed in L (i.e., $k = \hat k$ where $\hat k$ is the set of elements in L algebraic over k) and L is separable over k, or equivalently, $L \otimes_k \overline{k}$ is an integral domain when $\overline{k}$ is the algebraic closure of $k$ (that is, to say, $L, \overline{k}$ are linearly disjoint over k).Fried & Jarden (2008) p.38Cohn (2003) p.425

Properties

Regularity is transitive: if F/E and E/K are regular then so is F/K.Fried & Jarden (2008) p.39
If F/K is regular then so is E/K for any E between F and K.
The extension L/k is regular if and only if every subfield of L finitely generated over k is regular over k.
Any extension of an algebraically closed field is regular.Cohn (2003) p.426
An extension is regular if and only if it is separable and primary.Fried & Jarden (2008) p.44
A purely transcendental extension of a field is regular.

Self-regular extension

There is also a similar notion: a field extension $L / k$ is said to be self-regular if $L \otimes_k L$ is an integral domain. A self-regular extension is relatively algebraically closed in k.Cohn (2003) p.427 However, a self-regular extension is not necessarily regular.{{Citation needed|date=February 2010}}

References

{{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 extensions