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 $\overline {k}$ is the algebraic closure of $k$ (that is, to say, ${\displaystyle L,{\overline {k}}}$ 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 ${\displaystyle L/k}$ 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.^[6] However, a self-regular extension is not necessarily regular.

References

1. Fried & Jarden (2008) p.38
2. Cohn (2003) p.425
3. Fried & Jarden (2008) p.39
4. Cohn (2003) p.426
5. Fried & Jarden (2008) p.44
6. Cohn (2003) p.427

• 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. 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.