TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

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

The elements separable over the base form an intermediate field

Statement

For an algebraic extension $K / F$ , the set

$K_{s} : = {a \in K : a is separable over F}$

is an intermediate field between $F$ and $K$ .

Facts & Assumptions

Given: An algebraic extension $K / F$ and separable elements $a, b \in K$ .

[L1]

An algebraic extension generated by separable elements is separable (An algebraic extension generated by separable elements is separable).

[L2]

The subfield criterion requires $0, 1$ , closure under subtraction and multiplication, and inverses of nonzero elements (Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations).

Proof

technique · direct

1.1L1

The extension $F (a, b) / F$ is generated by separable elements, so [L1] makes every element of $F (a, b)$ separable over $F$ .

2.1step 1.1algebra

In particular, $a - b$ and $a b$ are separable, and if $a \neq 0$ then $a^{- 1}$ is separable. The elements $0$ and $1$ lie in $F$ and have linear minimal polynomials, so they are separable.

3.1step 2.1L2∎

The set $K_{s}$ therefore satisfies the subfield criterion [L2] and contains $F$ , so it is an intermediate field.

Depends on

Used by

The separable closure of the base inside an algebraic extension Definition

Dependency tree · two levels

18 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)