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Difference between revisions of "Purely inseparable extension"

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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 $a \in K$.
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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. 
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The purely inseparable extensions of a field $k$ form a [[distinguished class of extensions]].
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The ''exponent of a purely inseparable extension'' $L/K$ is the minimum $e$, if it exists, such that $L^{p^e} \subseteq K$.
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See also: [[Separable extension]]
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====References====
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* 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}}

Latest revision as of 09:02, 26 November 2023

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=39685