ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck 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 new irreducible denominator stays outside a finitely generated subalgebra of k(t)

Example

In the subalgebra $B = k [t, 1 / t] \subseteq k (t)$ , the element $1 / (t - 1)$ does not belong to $B$ .

Facts & Assumptions

Given: A field $k$ , the rational function field $k (t)$ , and the subalgebra $B = k [t, 1 / t]$ .

[L1]

The one-variable denominator obstruction says that finitely many allowed denominator factors cannot generate all of $k (t)$ over $k [t]$ (The rational function field k(t) is not finite over k[t]).

Verification

technique · direct

1.1

Every element of $B$ can be written as $h / t^{m}$ with $h \in k [t]$ and $m \geq 0$ . Thus only powers of $t$ occur in the denominator.

given

2.1

If $1 / (t - 1)$ belonged to $B$ , then $1 / (t - 1) = h / t^{m}$ for some $h \in k [t]$ . Cross-multiplication would give $t^{m} = h (t - 1)$ , so $t - 1$ would divide $t^{m}$ . But substituting $t = 1$ gives $1 = 0$ , impossible. This is the concrete obstruction behind [L1].

L1step 1.1∎

Depends on

The rational function field k(t) is not finite over k[t]

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, A Primer of Commutative Algebra, v4.03, Theorem 13.1 (standard reference, not scraped)