Primary extension

In field theory, a branch of algebra, a primary extension L of K is a field extension such that the algebraic closure of K in L is purely inseparable over K.^[1]

Properties

• An extension L/K is primary if and only if it is linearly disjoint from the separable closure of K over K.^[1]
• A subextension of a primary extension is primary.^[1]
• A primary extension of a primary extension is primary (transitivity).^[1]
• Any extension of a separably closed field is primary.^[1]
• An extension is regular if and only if it is separable and primary.^[1]
• A primary extension of a perfect field is regular.

References

• Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 11 (3rd revised ed.). Springer-Verlag. pp. 38–44. ISBN 978-3-540-77269-9. Zbl 1145.12001.