In
field theory, a branch of algebra, a primary extension ''L'' of ''K'' is a
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
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 In mathematics, algebras ''A'', ''B'' over a field ''k'' inside some field extension \Omega of ''k'' are said to be linearly disjoint over ''k'' if the following equivalent conditions are met:
*(i) The map A \otimes_k B \to AB induced by (x, y) \ma ...
from the
separable closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
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
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
is primary.[
* An extension is regular if and only if it is separable and primary.][
* A primary extension of a ]perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds:
* Every irreducible polynomial over ''k'' has distinct roots.
* Every irreducible polynomial over ''k'' is separable.
* Every finite extension of ''k'' is ...
is regular.
References
*
Field (mathematics)
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