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''.<ref name=FJ44>Fried & Jarden (2008) p.44</ref>
==Properties== * An extension ''L''/''K'' is primary if and only if it is linearly disjoint from the separable closure of ''K'' over ''K''.<ref name=FJ44/> * A subextension of a primary extension is primary.<ref name=FJ44/> * A primary extension of a primary extension is primary (transitivity).<ref name=FJ44/> * Any extension of a separably closed field is primary.<ref name=FJ44/> * An extension is regular if and only if it is separable and primary.<ref name=FJ44/> * A primary extension of a perfect field is regular.
==References== {{reflist}} * {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe |author-link=Michael D. Fried |author-link2=Moshe Jarden | 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 }}
Category:Field extensions
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