In field theory, a branch of algebra, a field extension \( {\displaystyle L/k} \) is said to be regular if k is algebraically closed in L (i.e., \( {\displaystyle k={\hat {k}}} \) where \( {\displaystyle {\hat {k}}} \) is the set of elements in L algebraic over k) and L is separable over k, or equivalently, \( {\displaystyle L\otimes _{k}{\overline {k}}} \) is an integral domain when \( \overline {k} \) is the algebraic closure of k (that is, to say, L , \( {\displaystyle L,{\overline {k}}} \) 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 \( {\displaystyle L/k} \) is said to be self-regular if \( {\displaystyle L\otimes _{k}L} \) is an integral domain. A self-regular extension is relatively algebraically closed in k.[6] However, a self-regular extension is not necessarily regular.[
References
Fried & Jarden (2008) p.38
Cohn (2003) p.425
Fried & Jarden (2008) p.39
Cohn (2003) p.426
Fried & Jarden (2008) p.44
Cohn (2003) p.427
Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 11 (3rd revised ed.). Springer-Verlag. pp. 38–41. ISBN 978-3-540-77269-9. Zbl 1145.12001.
M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese) [1]
Cohn, P. M. (2003). Basic Algebra. Groups, Rings, and Fields. Springer-Verlag. ISBN 1-85233-587-4. Zbl 1003.00001.
A. Weil, Foundations of algebraic geometry.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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