Regular extension
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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).
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.
References
M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese) A. Weil, Foundations of algebraic geometry.
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