Regular extension

In field theory, a branch of algebra, a field extension L / k {displaystyle L/k} is said to be regular if k is algebraically closed in L (i.e., k = k ^ {displaystyle k={hat {k}}} where k ^ {displaystyle {hat {k}}} is the set of elements in L algebraic over k) and L is separable over k, or equivalently, L ⊗ k k ¯ {displaystyle Lotimes _{k}{overline {k}}} is an integral domain when k ¯ {displaystyle {overline {k}}} is the algebraic closure of k {displaystyle k} (that is, to say, L , k ¯ {displaystyle L,{overline {k}}} are linearly disjoint over k). In field theory, a branch of algebra, a field extension L / k {displaystyle L/k} is said to be regular if k is algebraically closed in L (i.e., k = k ^ {displaystyle k={hat {k}}} where k ^ {displaystyle {hat {k}}} is the set of elements in L algebraic over k) and L is separable over k, or equivalently, L ⊗ k k ¯ {displaystyle Lotimes _{k}{overline {k}}} is an integral domain when k ¯ {displaystyle {overline {k}}} is the algebraic closure of k {displaystyle k} (that is, to say, L , k ¯ {displaystyle L,{overline {k}}} are linearly disjoint over k). There is also a similar notion: a field extension L / k {displaystyle L/k} is said to be self-regular if L ⊗ k L {displaystyle Lotimes _{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.