Namespaces
Variants
Actions

Purely inseparable extension

From Encyclopedia of Mathematics
Revision as of 20:27, 8 November 2016 by Richard Pinch (talk | contribs) (Start article: Purely inseparable extension)
Jump to: navigation, search

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