Regular extension
In field theory, a branch of algebra, a field extension [math]\displaystyle{ L/k }[/math] is said to be regular if k is algebraically closed in L (i.e., [math]\displaystyle{ k = \hat k }[/math] where [math]\displaystyle{ \hat k }[/math] is the set of elements in L algebraic over k) and L is separable over k, or equivalently, [math]\displaystyle{ L \otimes_k \overline{k} }[/math] is an integral domain when [math]\displaystyle{ \overline{k} }[/math] is the algebraic closure of [math]\displaystyle{ k }[/math] (that is, to say, [math]\displaystyle{ L, \overline{k} }[/math] are linearly disjoint over k).[1][2]
Properties
- Regularity is transitive: if F/E and E/K are regular then so is F/K.[3]
- If F/K is regular then so is E/K for any E between F and K.[3]
- The extension L/k is regular if and only if every subfield of L finitely generated over k is regular over k.[2]
- Any extension of an algebraically closed field is regular.[3][4]
- An extension is regular if and only if it is separable and primary.[5]
- A purely transcendental extension of a field is regular.
Self-regular extension
There is also a similar notion: a field extension [math]\displaystyle{ L / k }[/math] is said to be self-regular if [math]\displaystyle{ L \otimes_k L }[/math] is an integral domain. A self-regular extension is relatively algebraically closed in k.[6] However, a self-regular extension is not necessarily regular.[citation needed]
References
- Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 11 (3rd revised ed.). Springer-Verlag. pp. 38–41. ISBN 978-3-540-77269-9.
- 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.
- A. Weil, Foundations of algebraic geometry.
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