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).

• 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 ${\displaystyle L/k}$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. However, a self-regular extension is not necessarily regular.

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• M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese) [1]
• Cohn, P. M. (2003). Basic Algebra. Groups, Rings, and Fields. Springer-Verlag. ISBN 1-85233-587-4. Zbl 1003.00001.
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