View source for Purely inseparable extension

{{TEX|done}}{{MSC|12F15}}

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 $L^{p^e} \subseteq K$.

See also: [[Separable extension]]

====References====
* N. Jacobson, "Lectures in Abstract Algebra: III. Theory of Fields and Galois Theory"  Graduate Texts in Mathematics '''32''' Springer (1980) {{ISBN|0-387-90124-8}} {{ZBL|0455.12001}}