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.

An extension that is both separable and purely inseparable is trivial

Statement

If an algebraic extension $K / F$ is both separable and purely inseparable, then $K = F$ .

Facts & Assumptions

Given: An algebraic extension $K / F$ that is separable and purely inseparable, and an element $α \in K$ .

[L1]

Separability makes the minimal polynomial of every element separable (Separable algebraic elements and separable extensions).

[L2]

Pure inseparability makes every element have exactly one distinct conjugate over the base (Pure inseparability and its conjugate, embedding, and separable-degree criteria).

[L3]

Every algebraic element has a monic irreducible minimal polynomial over the base (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element).

Proof

technique · direct

1.1L1L2L3

Let $m_{α}$ be the minimal polynomial from [L3]. It is separable by [L1], so all of its roots are distinct, but [L2] says it has only one distinct root. Hence $\deg m_{α} = 1$ .

2.1step 1.1algebra∎

A degree-one minimal polynomial puts $α$ in $F$ . Since $α \in K$ was arbitrary, $K = F$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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)