CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-17

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A finite purely inseparable extension in characteristic $p$ has degree a power of $p$

Statement

If $K / F$ is a finite purely inseparable extension of characteristic $p > 0$ , then $[K : F] = p^{r}$ for some $r \in N$ . The trivial extension gives $r = 0$ .

Facts & Assumptions

Given: A finite purely inseparable extension $K / F$ of characteristic $p > 0$ .

[L1]

The minimal polynomial of each element has the form $x^{p^{e}} - a$ and hence has $p$ -power degree (Pure inseparability and its conjugate, embedding, and separable-degree criteria).

[L2]

The degree of a simple algebraic extension is the degree of its minimal polynomial (A simple algebraic extension is its minimal-polynomial quotient and has power basis $1, a, \dots, a^{n - 1}$ and degree $n$ ).

[L3]

Degrees multiply in a finite tower (Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ ).

[L4]

A finite extension has a finite basis over its base (The degree $[K : F] = \dim_{F} K$ of a finite field extension).

Proof

technique · direct

1.1L4choose

Choose a finite basis using [L4]; its elements generate $K$ , so adjoining them successively gives a finite tower of simple extensions.

2.1step 1.1L1L2L3

At each nontrivial step, [L1] and [L2] make the degree a power of $p$ . The tower law [L3] makes the product, and hence $[K : F]$ , a power of $p$ .

3.1L3∎

If $K = F$ , the empty tower has degree $1 = p^{0}$ , so the boundary case is included.

Depends on

Used by

[K:F]=[K:F]ₛ[K:F]ᵢ, and in positive characteristic the inseparable degree is a power of p Corollary

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

P. L. Clark, Field Theory, Chapters 4 and 5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapters 3 and 5 (standard reference, not scraped)

A finite purely inseparable extension in characteristic pp has degree a power of pp