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).¹²

• Regularity is transitive: if F/E and E/K are regular then so is F/K.³
• If F/K is regular then so is E/K for any E between F and K.³
• The extension L/k is regular if and only if every subfield of L finitely generated over k is regular over k.²
• Any extension of an algebraically closed field is regular.³⁴
• An extension is regular if and only if it is separable and primary.⁵
• A purely transcendental extension of a field is regular.

There is also a similar notion: a field extension ${\displaystyle L/k}$ is said to be self-regular if ${\displaystyle L\otimes _{k}L}$ is an integral domain. A self-regular extension is relatively algebraically closed in k.⁶ However, a self-regular extension is not necessarily regular.