Regular extension - Misplaced Pages

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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, $L\otimes _{k}{\overline {k}}$ is an integral domain when ${\overline {k}}$ is the algebraic closure of $k$ (that is, to say, $L,{\overline {k}}$ are linearly disjoint over k).

Properties

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.

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. However, a self-regular extension is not necessarily regular.

References

Fried & Jarden (2008) p.38
^ Cohn (2003) p.425
^ Fried & Jarden (2008) p.39
Cohn (2003) p.426
Fried & Jarden (2008) p.44
Cohn (2003) p.427

Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. pp. 38–41. ISBN 978-3-540-77269-9. Zbl 1145.12001.
M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese)
Cohn, P. M. (2003). Basic Algebra. Groups, Rings, and Fields. Springer-Verlag. ISBN 1-85233-587-4. Zbl 1003.00001.
A. Weil, Foundations of algebraic geometry.

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