TheoremStatement: 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.

Ramification and residue degrees in towers

Statement

For $M / L / K$ and $Q ∣ P ∣ p$ , $e (Q / p) = e (Q / P) e (P / p), f (Q / p) = f (Q / P) f (P / p) .$

Proof

Given: the indicated tower and primes.

1.1

Substitute the prime factorisation of $p O_{L}$ into its extension to $O_{M}$ and compare the exponent of $Q$ .

givenalgebra

2.1

The three residue fields form a finite tower, so the dimensions multiply. These are exactly the two asserted equalities.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, Algebraic Number Theory, Chapter 4, Exercise 4-2 (standard reference, not scraped)