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, $L \otimes_k \overline$ is an integral domain when $\overline$ is the algebraic closure of $k$ (that is, to say, $L, \overline$ are linearly disjoint over ''k'').Fried & Jarden (2008) p.38Cohn (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.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 $L / k$ 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''.Cohn (2003) p.427 However, a self-regular extension is not necessarily regular.

References

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* M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese

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* A. Weil, Foundations of algebraic geometry.

Field (mathematics)

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