# Regular extension

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In [field theory](/source/Field_theory_(mathematics)), a branch of algebra, a [field extension](/source/Field_extension) L/k is said to be **regular** if *k* is [algebraically closed](/source/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](/source/Separable_extension) 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](/source/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](/source/Primary_extension).[5]
- A [purely transcendental extension](/source/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*.[6] However, a self-regular extension is not necessarily regular.[citation needed]

## References

1. Fried & Jarden (2008) p.38

1. Cohn (2003) p.425

1. Fried & Jarden (2008) p.39

1. Cohn (2003) p.426

1. Fried & Jarden (2008) p.44

1. Cohn (2003) p.427

- Fried, Michael D. & Jarden, Moshe (2008). *Field arithmetic*. Vol. 11. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 3rd revised ed. [Springer-Verlag](/source/Springer-Verlag). pp. 38–41. ISBN 978-3-540-77269-9. Zbl 1145.12001.
- M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese) [\[1\]](http://www.shokabo.co.jp/mybooks/ISBN978-4-7853-1309-8.htm)
- Cohn, P. M. (2003). *Basic Algebra. Groups, Rings, and Fields*. [Springer-Verlag](/source/Springer-Verlag). ISBN 1-85233-587-4. Zbl 1003.00001.
- A. Weil, [Foundations of algebraic geometry](/source/Foundations_of_algebraic_geometry).

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