Purely inseparable extension

In algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of an equation of the form x^q = a, with q a power of p and a in k. Purely inseparable extensions are sometimes called radicial extensions, which should not be confused with the similar-sounding but more general notion of radical extensions.

An algebraic extension ${\displaystyle E\supseteq F}$ is a purely inseparable extension if and only if for every ${\displaystyle \alpha \in E\setminus F}$, the minimal polynomial of ${\displaystyle \alpha }$ over F is not a separable polynomial. If F is any field, the trivial extension ${\displaystyle F\supseteq F}$ is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section.

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