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).

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 $$L / k$$ 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. However, a self-regular extension is not necessarily regular.