# Purely inseparable extension

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2010 Mathematics Subject Classification: Primary: 12F15 [MSN][ZBL]

A field extension $L/K$ in finite characteristic $p$ in which every element of $L$ which is algebraic over $K$ is a purely inseparable element: that is, has a minimal polynomial of the form $X^{p^e} - a$ where $a \in K$.

Let $E/K$ be an arbitrary algebraic extension. The elements of the field $E$ that are separable over $K$ form a field, $S$, which is the maximal separable extension of $K$ contained in $E$. Then $S/K$ is a separable extension and $E/S$ is a purely inseparable extension.

The purely inseparable extensions of a field $k$ form a distinguished class of extensions.

The exponent of a purely inseparable extension $L/K$ is the minimum $e$, if it exists, such that $L^{p^e} \subseteq K$.