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.

The separable degree of $F (α) / F$ is the number of distinct roots of $m_{α}$

Statement

If $α$ is algebraic over $F$ , then $[F (α) : F]_{s}$ equals the number of distinct roots of the minimal polynomial $m_{α}$ in any algebraic closure of $F$ .

Facts & Assumptions

Given: An algebraic element $α$ over $F$ and an algebraic closure $Ω / F$ .

[L1]

Separable degree counts $F$ -embeddings into an algebraic closure (The separable degree $[K : F]_{s}$ as a count of embeddings into an algebraic closure).

[L2]

Such embeddings of $F (α)$ correspond bijectively to the distinct roots of $m_{α}$ ( $F$ -embeddings of $F (α)$ into an algebraically closed field correspond to the distinct roots of $m_{α}$ ).

[L3]

The embedding count is independent of the chosen algebraic closure (The separable degree is independent of the chosen algebraic closure).

Proof

technique · direct

1.1L1L2

By [L2], the embedding set counted in [L1] is in bijection with the distinct-root set of $m_{α}$ in $Ω$ .

2.1step 1.1L3∎

Taking finite cardinalities gives the assertion, and [L3] removes dependence on $Ω$ .

Depends on

Used by

Dependency tree · two levels

14 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)

The separable degree of F(α)/FF(α)/F is the number of distinct roots of mαmα​