Purely inseparable extension

This article defines a field extension property: a property that can be evaluated to true/false for any field extension.
View a complete list of field extension properties|View a complete list of field properties

Definition

Suppose $L$ is a field extension of a field $K$. We say that $L$ is purely inseparable over $K$ if it satisfies the following equivalent conditions:

1. The extension is a normal extension and its automorphism group is trivial.
2. $L$ is an algebraic extension of $K$ such that if $M$ is a normal extension of $K$ containing $L$, the fixed field of $\operatorname {Aut} (M/K)$ contains $L$.
3. The minimal polynomial of any element of $L$ over $K$ is a power of a linear polynomial.
4. If $K$ has characteristic zero, $K=L$. If $K$ has characteristic $p$, every element $a\in L$ satisfies $a^{p^{n}}\in K$ for some $n$ (depending on $a$).

Relation with other properties

Weaker properties

• Normal extension
• Algebraic extension

Opposite properties

• Galois extension is a normal extension that is also a separable extension.