# Regular extension

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.[citation needed]