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).Fried & Jarden (2008) p.38Cohn (2003) p.425
Properties Regularity is transitive: if F/E and E/K are regular then so is F/K.Fried & Jarden (2008) p.39 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.Cohn (2003) p.426 An extension is regular if and only if it is separable and primary.Fried & Jarden (2008) p.44 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.Cohn (2003) p.427 However, a self-regular extension is not necessarily regular.
References
M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese)
A. Weil, Foundations of algebraic geometry.