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.Fried & Jarden (2008) p.44
Properties
- An extension L/K is primary if and only if it is linearly disjoint from the separable closure of K over K.
- A subextension of a primary extension is primary.
- A primary extension of a primary extension is primary (transitivity).
- Any extension of a separably closed field is primary.
- An extension is regular if and only if it is separable and primary.
- A primary extension of a perfect field is regular.
References
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- {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd revised | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=Springer-Verlag | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 | pages=38–44 }}
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