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purely inseparable (Definition)

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 = \mathbb{F}_3(s)$ where $ \mathbb{F}_3$ is the finite field with $ 3$ elements. Let $ F = \mathbb{F}_3(s^6)$. Then $ K/F$ is neither separable, nor purely inseparable. Let $ E = \mathbb{F}_3(s^3)$. Then $ E/F$ is separable, and $ K/E$ is purely inseparable.



"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
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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 1636 times total.

Classification:
AMS MSC: 12F15 (Field theory and polynomials :: Field extensions :: Inseparable extensions)
Pending Errata and Addenda
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