Regular extension

In field theory, a branch of algebra, a field extension ${\displaystyle L/k}$ is said to be regular if k is algebraically closed in L (i.e., ${\displaystyle k={\hat {k}}}$ where ${\displaystyle {\hat {k}}}$ is the set of elements in L algebraic over k) and L is separable over k, or equivalently, ${\displaystyle L\otimes _{k}{\overline {k}}}$ is an integral domain when ${\displaystyle {\overline {k}}}$ is the algebraic closure of ${\displaystyle k}$ (that is, to say, ${\displaystyle L,{\overline {k}}}$ are linearly disjoint over k).^[1][2]