# 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 & Jarden (2008) p.44

- 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.

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