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). 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. 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.