Regular extension

In field theory, a branch of algebra, a field extension is said to be regular if k is algebraically closed in L and L is separable over k, or equivalently, is an integral domain when is the algebraic closure of .

Properties

Regularity is transitive: if F/''E and E''/K are regular then so is F/''K.
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.
An extension is regular if and only if it is separable and primary.
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. However, a self-regular extension is not necessarily regular.