TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedaudited 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 discriminant is the norm of the different

Statement

$N D_{K} = ∣ d_{K} ∣$ .

Proof

Given: an integral basis and its trace Gram matrix $G$ .

1.1givenalgebra

The dual lattice is obtained from the original basis by the inverse matrix $G^{- 1}$ , hence its index relative to $O_{K}$ has absolute determinant $∣ \det G ∣$ .

2.1step 1.1algebra∎

Since $\det G = d_{K}$ and the different is the inverse codifferent, its norm is that same positive index.

Depends on

Used by

Discriminant valuations from different exponents Corollary
A quadratic-field codifferent Example

Dependency tree · two levels

10 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, Theorem 4.8 (standard reference, not scraped)