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 = k ^ where k ^ is the set of elements in L algebraic over k) and L is separable over k, or equivalently, L ⊗ k k ¯ is an integral domain when k ¯ is the algebraic closure of k (that is, to say, L , k ¯ 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 L / k is said to be self-regular if L ⊗ 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.
