LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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 codifferent is a fractional ideal

Statement

For every nonzero fractional ideal $A$ of $O_{K}$ , $A^{\lor}$ is a fractional ideal and $(c A)^{\lor} = c^{- 1} A^{\lor}$ for $c \in K^{\times}$ .

Proof

Given: a fractional ideal $A$ and $c \in K^{\times}$ .

1.1

In a $Z$ -basis of the full lattice $A$ , nondegeneracy of trace gives a dual basis, so $A^{\lor}$ is again a full lattice and is stable under $O_{K}$ .

givenalgebra

2.1

The condition $Tr (x c A) \subseteq Z$ is equivalent to $c x \in A^{\lor}$ , proving $(c A)^{\lor} = c^{- 1} A^{\lor}$ and the fractional-ideal claim.

step 1.1algebra∎

Depends on

Used by

The different of a number field Definition

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

Keith Conrad, The Different Ideal, Theorems 3.4 and 3.9 (standard reference, not scraped)