In
field theory, a branch of algebra, a
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
is said to be regular if ''k'' is
algebraically closed in ''L'' (i.e.,
where
is the set of elements in ''L'' algebraic over ''k'') and ''L'' is
separable over ''k'', or equivalently,
is an integral domain when
is the algebraic closure of
(that is, to say,
are
linearly disjoint In mathematics, algebras ''A'', ''B'' over a field ''k'' inside some field extension \Omega of ''k'' are said to be linearly disjoint over ''k'' if the following equivalent conditions are met:
*(i) The map A \otimes_k B \to AB induced by (x, y) \m ...
over ''k'').
[Fried & Jarden (2008) p.38][Cohn (2003) p.425]
Properties
* Regularity is transitive: if ''F''/''E'' and ''E''/''K'' are regular then so is ''F''/''K''.
[Fried & Jarden (2008) p.39]
* 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.][Cohn (2003) p.426]
* An extension is regular if and only if it is separable and primary
Primary or primaries may refer to:
Arts, entertainment, and media Music Groups and labels
* Primary (band), from Australia
* Primary (musician), hip hop musician and record producer from South Korea
* Primary Music, Israeli record label
Work ...
.[Fried & Jarden (2008) p.44]
* A purely transcendental extension of a field is regular.
Self-regular extension
There is also a similar notion: a field extension is said to be self-regular if is an integral domain. A self-regular extension is relatively algebraically closed in ''k''.[Cohn (2003) p.427] However, a self-regular extension is not necessarily regular.
References
*
* M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese
*
* A. Weil, Foundations of algebraic geometry
''Foundations of Algebraic Geometry'' is a book by that develops algebraic geometry over fields of any characteristic. In particular it gives a careful treatment of intersection theory by defining the local intersection multiplicity of two subv ...
.
Field (mathematics)
{{Abstract-algebra-stub