CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

For a finite extension, $[K : F]_{s} \leq [K : F]$

Statement

For every finite field extension $K / F$ , one has $[K : F]_{s} \leq [K : F]$ .

Facts & Assumptions

Given: A finite extension $K / F$ .

[L1]

Separable degree is multiplicative in finite towers (Separable degree is multiplicative in finite towers: $[L : F]_{s} = [L : K]_{s} [K : F]_{s}$ ).

[L2]

Embeddings of a simple algebraic extension correspond to the distinct roots of its minimal polynomial ( $F$ -embeddings of $F (α)$ into an algebraically closed field correspond to the distinct roots of $m_{α}$ ).

[L3]

Ordinary extension degrees multiply in finite towers (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 $F$ -basis of $K$ by [L4]; its elements generate $K$ , so adjoining them successively gives a finite tower of simple extensions from $F$ to $K$ .

1.2L2

At each simple step, [L2] counts embeddings by distinct roots of a minimal polynomial, so its separable degree is at most the degree of that polynomial, which is the ordinary degree of the step.

2.1step 1.1step 1.2L1L3algebra∎

Multiplying the inequalities in step 1.2 and using [L1] and [L3] for the two tower products gives $[K : F]_{s} \leq [K : F]$ .

Depends on

Used by

Dependency tree · two levels

12 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)

For a finite extension, [K:F]s≤[K:F][K:F]s​≤[K:F]