TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Clearing denominators for an algebraic number

Statement

For every $α \in K$ , some positive integer $m$ satisfies $m α \in O_{K}$ ; hence $K = Frac (O_{K})$ .

Facts & Assumptions

Given: $α \in K$ .

Proof

technique · direct

1.1

Write the monic minimal polynomial of $α$ as $X^{n} + c_{n - 1} X^{n - 1} + \dots + c_{0} \in Q [X],$ and choose a positive common denominator $m$ of the $c_{i}$ . Multiplying the equation for $α$ by $m^{n}$ shows that $m α$ satisfies $X^{n} + (m c_{n - 1}) X^{n - 1} + (m^{2} c_{n - 2}) X^{n - 2} + \dots + m^{n} c_{0},$ a monic polynomial in $Z [X]$ .

givenalgebra

2.1

Thus $m α \in O_{K}$ , and $α = (m α) / m$ lies in its fraction field; the reverse inclusion is contained in $K$ .

step 1.1∎

Depends on

Used by

The ring of integers has rank the degree Theorem

Dependency tree · two levels

4 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

Milne, Proposition 2.6 (standard reference, not scraped)