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purely inseparable
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(Definition)
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Let $F$ be a field of characteristic $p > 0$ and let $\alpha$ be an element which is algebraic over $F$ . Then $\alpha$ is said to be purely inseparable over $F$ if $\alpha^{p^n} \in F$ for some $n \ge 0$ .
An algebraic field extension $K/F$ is purely inseparable if each element of $K$ is purely inseparable over $F$ .
Purely inseparable extensions have the following property: if $K/F$ is purely inseparable, and $A$ is an algebraic closure of $F$ which contains $K$ , then any homomorphism $K \to A$ which fixes $F$ necessarily fixes $K$ .
Let $K/F$ be an arbitrary algebraic extension. Then there is an intermediate field $E$ such that $K/E$ is purely inseparable, and $E/F$ is separable.
Example 1 Let $s$ be an indeterminate, and let $K = \F_3(s)$ where $\F_3$ is the finite field with $3$ elements. Let $F = \F_3(s^6)$ . Then $K/F$ is neither separable, nor purely inseparable. Let $E = \F_3(s^3)$ . Then $E/F$ is separable, and $K/E$ is purely inseparable.
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"purely inseparable" is owned by mclase.
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Cross-references: finite field, indeterminate, separable, homomorphism, contains, algebraic closure, property, extensions, algebraic field extension, algebraic, characteristic, field
There are 4 references to this entry.
This is version 3 of purely inseparable, born on 2004-11-16, modified 2006-10-18.
Object id is 6480, canonical name is PurelyInseparable.
Accessed 2487 times total.
Classification:
| AMS MSC: | 12F15 (Field theory and polynomials :: Field extensions :: Inseparable extensions) |
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Pending Errata and Addenda
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