# Regular extension

In field theory, a branch of algebra, a field extension is said to be **regular** if *k* is algebraically closed in *L* (i.e., where is the set of elements in *L* algebraic over *k*) 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.

## References

- 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)
- 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.

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