Primary extension explained

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

• Book: Fried . Michael D. . Jarden . Moshe . Field arithmetic . 3rd revised . Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge . 11 . . 2008 . 978-3-540-77269-9 . 1145.12001 . 38–44 .

Notes and References

1. Fried & Jarden (2008) p.44