CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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

Statement

For every finite extension $K / F$ ,

$[K : F] = [K : F]_{s} [K : F]_{i} .$

If $char F = p > 0$ , then $[K : F]_{i}$ is a power of $p$ . In characteristic zero it is one.

Facts & Assumptions

Given: A finite extension $K / F$ with relative separable closure $K_{s}$ .

[L1]

Inseparable degree is the quotient $[K : F] / [K : F]_{s}$ (The inseparable degree $[K : F]_{i} = [K : F] / [K : F]_{s}$ of a finite extension).

[L2]

One has $[K : F]_{s} = [K_{s} : F]$ (For a finite extension, $[K : F]_{s} = [K_{s} : F]$ ).

[L3]

The extension $K / K_{s}$ is purely inseparable (An algebraic extension is purely inseparable over its separable closure).

[L4]

A finite purely inseparable extension in characteristic $p$ has $p$ -power degree (A finite purely inseparable extension in characteristic $p$ has degree a power of $p$ ).

[L5]

Ordinary degrees multiply in finite towers (Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ ).

Proof

technique · direct

1.1

The displayed factorization is the defining equality in [L1] after multiplying by $[K : F]_{s}$ .

L1algebra

2.1

By [L5] and [L2], $[K : F] = [K : K_{s}] [K_{s} : F] = [K : K_{s}] [K : F]_{s}$ , so comparison with step 1.1 gives $[K : F]_{i} = [K : K_{s}]$ .

L2L5algebra

3.1

In characteristic $p > 0$ , [L3] and [L4] make this last degree a power of $p$ . In characteristic zero, $K_{s} = K$ , so it is one.

step 2.1L3L4∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 41 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)

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