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.