Definition:Purely Inseparable Field Extension
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Definition
Let $E/F$ be an algebraic field extension.
Definition 1
The extension $E/F$ is purely inseparable if and only if every element $\alpha \in E \setminus F$ is inseparable.
Definition 2
Let $F$ have positive characteristic $p$.
The extension $E/F$ is purely inseparable if and only if for each $\alpha \in E$ there exists $n \in \N$ such that $\alpha^{p^n} \in F$.
Definition 3
Let $F$ have positive characteristic $p$.
The extension $E/F$ is purely inseparable if and only if each element of $E$ has a minimal polynomial of the form $X^{p^n} - a$.
Also see
- Equivalence of Definitions of Purely Inseparable Extension
- Definition:Relative Purely Inseparable Closure
- Definition:Purely Inseparable Closure
- Definition:Separable Field Extension
- Definition:Inseparable Field Extension
Sources
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- 1994: I. Martin Isaacs: Algebra: A Graduate Course: Chapter $19$ Separability and Inseparability $\S19$B
- This article incorporates material from purely inseparable on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
- Weisstein, Eric W. "Purely Inseparable Extension." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PurelyInseparableExtension.html