ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30

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.

A transcendence basis of k(s, t, sqrt(s+t)) over k

Example

Let $K = k (s, t, \sqrt{s + t})$ , where $s$ and $t$ are algebraically independent over $k$ . Then ${s, t}$ is a transcendence basis of $K$ over $k$ .

Facts & Assumptions

Given: A field $k$ , algebraically independent elements $s, t$ over $k$ , and $u = \sqrt{s + t}$ .

[L1]

An algebraically independent set is a transcendence basis once the ambient field is algebraic over the generated field (A maximal algebraically independent set is a transcendence basis).

Verification

technique · direct

1.1

The set ${s, t}$ is algebraically independent over $k$ by construction, so $k (s, t)$ is a rational function field in two variables.

given

2.1

The remaining generator $u$ satisfies the polynomial $X^{2} - (s + t) \in k (s, t) [X]$ , so $u$ is algebraic over $k (s, t)$ . Since $K = k (s, t, u)$ , the whole field $K$ is algebraic over $k (s, t)$ .

step 1.1given

3.1

Therefore [L1] shows that ${s, t}$ is a transcendence basis of $K$ over $k$ .

L1step 2.1∎

Depends on

A maximal algebraically independent set is a transcendence basis

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

J. S. Milne, Fields and Galois Theory, v5.10, Chapter 9 (standard reference, not scraped)