# Purely inseparable extension

2020 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 $L^{p^e} \subseteq K$.

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

**How to Cite This Entry:**

Purely inseparable extension.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Purely_inseparable_extension&oldid=54725