Article · Wikipedia archive · Last revised Jun 30, 2026

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.

Last revised

Jun 30, 2026

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≈ 1 min

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Source

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

References

Fried & Jarden (2008) p.44

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