Vipul: Created page with '{{field extension property}} ==Definition== Suppose $L$ is a field extension of a field $K$ . We say that $L$ is '''purely inseparable''...'

2009-05-14T23:16:24Z

Created page with '{{field extension property}} ==Definition== Suppose <math>L</math> is a field extension of a field <math>K</math>. We say that <math>L</math> is '''purely inseparable''...'

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{{field extension property}}

==Definition==

Suppose <math>L</math> is a [[field extension]] of a [[field]] <math>K</math>. We say that <math>L</math> is '''purely inseparable''' over <math>K</math> if it satisfies the following equivalent conditions:

# The extension is a [[defining ingredient::normal extension]] and its [[defining ingredient::field extension with trivial automorphism group|automorphism group is trivial]].
# <math>L</math> is an [[algebraic extension]] of <math>K</math> such that if <math>M</math> is a [[normal extension]] of <math>K</math> containing <math>L</math>, the fixed field of <math>\operatorname{Aut}(M/K)</math> contains <math>L</math>.
# The [[minimal polynomial]] of any element of <math>L</math> over <math>K</math> is a power of a linear polynomial.
# If <math>K</math> has characteristic zero, <math>K = L</math>. If <math>K</math> has characteristic <math>p</math>, every element <math>a \in L</math> satisfies <math>a^{p^n} \in K</math> for some <math>n</math> (depending on <math>a</math>).

==Relation with other properties==

===Weaker properties===

* [[Stronger than::Normal extension]]
* [[Stronger than::Algebraic extension]]

===Opposite properties===

* [[Galois extension]] is a normal extension that is also a [[separable extension]].

Purely inseparable extension - Revision history

Vipul: Created page with '{{field extension property}} ==Definition== Suppose L is a field extension of a field K. We say that L is '''purely inseparable''...'

Vipul: Created page with '{{field extension property}} ==Definition== Suppose $L$ is a field extension of a field $K$ . We say that $L$ is '''purely inseparable''...'