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
=
k
^
where
k
^
is the set of elements in L algebraic over k) and L is separable over k, or equivalently,
L
⊗
k
k
¯
is an integral domain when
k
¯
is the algebraic closure of
k
(that is, to say,
L
,
k
¯
are linearly disjoint over k).
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.
There is also a similar notion: a field extension
L
/
k
is said to be self-regular if
L
⊗
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.
